Unfolding of the unramified irregular singular generalized isomonodromic deformation
UUNFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZEDISOMONODROMIC DEFORMATION
MICHI-AKI INABA
Abstract.
We introduce an unfolded moduli space of connections, which is an algebraic relative modulispace of connections on complex smooth projective curves, whose generic fiber is a moduli space of regularsingular connections and whose special fiber is a moduli space of unramified irregular singular connections.On the moduli space of unramified irregular singular connections, there is a subbundle of the tangentbundle defining the generalized isomonodromic deformation produced by the Jimbo-Miwa-Ueno theory.On an analytic open subset of the unfolded moduli space of connections, we construct a non-canonicallift of this subbundle, which we call an unfolding of the unramified irregular singular generalized isomon-odromic deformation. Our construction of an unfolding of the unramified irregular singular generalizedisomonodromic deformation is not compatible with the asymptotic property in the unfolding theory estab-lished by Hurtubise, Lambert and Rousseau which gives unfolded Stokes matrices for an unfolded lineardifferential equation in a general framework.
Introduction
The intention of this paper is to produce a tool toward understanding the confluence phenomena connect-ing the regular singular isomonodromic deformation and the irregular singular generalized isomonodromicdeformation. In the case of connections on P , the regular singular isomonodromic deformation is theSchlesinger equation and the unramified irregular singular generalized isomonodromic deformation is theJimbo-Miwa-Ueno equation which is completely given in [21], [22], [23]. The most fundamental exampleof the confluence phenomena will be the confluence of the classical hypergeometric functions, though theirisomonodromic deformations may not be mentioned because of the rigidity. There are extended results in[24] and [25]. The next important example of the confluence phenomena will be the degeneration of Painlev´eequations, where the irregular singular generalized isomonodromic deformation arises when we take a limitof the regular singular isomonodromic deformation. Observation of confluence of Painlev´e equations via τ function is given in [20] and further study via confluent conformal blocks are given in [32]. There is anapproach via monodromy manifolds in [33] to the confluence of Painlev´e equations. In [26], a generalizationof the confluence phenomena to a general Schlesinger equation is given. An origin of confluence problems isgiven by Ramis in [36] and unfolding of Stokes data is one of the important problems. Studies of confluenceproblem from this viewpoint are done in [38], [42] and [10]. A general framework of unfolded Stokes data ofan unfolded linear differential equation is established by Hurtubise, Lambert and Rousseau in [14] and [15].In [28], confluence of unfolded Stokes data in rank two case is given explicitly. One of the key ideas in theunfolding theory by Hurtubise, Lambert and Rousseau in [14] and [15] is to adopt fundamental solutionswith an asymptotic property, which is estimated by a flow of the vector field v (cid:15) = p (cid:15) ( x ) ∂∂x , where p (cid:15) ( x ) = 0is a local unfolding equation. They construct unfolded Stokes matrices of a linear differential equationon P via connecting fundamental solutions with an asymptotic property around points in the unfoldingdivisor and that around ∞ . In order to reconstruct an unfolded linear differential equation, they consideranother regular singular point, whose monodromy reflects the analytic continuation along the ‘inner side’of the unfolded divisor. In [15], they introduce a delicate condition called the ‘compatibility condition’ inorder that the corresponding linear differential equation is a well-defined analytic family.The author’s early hope was to understand the unfolding theory by Hurtubise, Lambert and Rousseauin a moduli theoretic way. So we introduce in this paper an unfolded moduli space of connections, whosegeneric fiber is a moduli space of regular singular connections and whose special fiber is a moduli space ofunramified irregular singular connections.The Schlesinger type equation, or the regular singular isomonodromic deformation is defined on a familyof moduli spaces of regular singular connections on smooth projective curves. In order to get a good modulispace, we consider a parabolic structure to the given connection and the moduli space is constructed in [34], Mathematics Subject Classification. a r X i v : . [ m a t h . AG ] J u l MICHI-AKI INABA [1], [16] and [17], which is a smooth and quasi-projective moduli space. The algebraic moduli constructionis basically given by modifying the standard method by Simpson in [40], [41] or by Nitsure in [35]. In [16]and [17], we formulate the regular singular isomonodromic deformation and prove the geometric Painlev´eproperty of the isomonodromic deformation using the properness of the Riemann-Hilbert morphism. In [45],the moduli space of filtered local systems is introduced by Yamakawa and the Riemann-Hilbert isomorphismvia the idea by Simpson in [39] is given, from which we can also prove the geometric Painlev´e propertyof the isomonodromic deformation. Moduli theoretic descriptions of the regular singular isomonodromicdeformation are also given in [13], [11], [12], [4], [5] and [44]. We notice that we cannot forget the parabolicstructure for the precise formulation of the isomonodromic deformation given in [17, Proposition 8.1] onthe locus where the parabolic structure is not completely determined by the given connection. Let us recallthat the essential number of independent variables of the regular singular isomonodromic deformation is3 g − D , where D is the divisor consisting of all the regular singular points and g is the genus of basecurves.Moduli space of unramified irregular singular connections is constructed in [3] analytically and in [19]algebraically. The irregular singular generalized isomonodromic deformation from the moduli theoreticviewpoint is given in [6], [7], [9], [13], [37], [44], [8] and [19] from various viewpoints, respectively. In spiteof the importance of parabolic structure in the regular singular case, unfolding problem of the modulispace of irregular singular connections does not seem to work well with parabolic structure, especially forthe deformation argument of ramified connections in [18, Theorem 4.1]. So we adopt another method ofparameterizing the local exponents in this paper. If we fix distinct complex numbers µ , . . . , µ r and if we takegeneric unramified local exponents ν dzz m , . . . , ν r dzz m at a singular point p , then we can observe that there isa polynomial ν ( T ) ∈ C [ z ] / ( z m )[ T ] satisfying ν k = ν ( µ k ) for any k . So we can regard ( ν ( T ) , µ , . . . , µ r ) asa data of local exponents. We can see that a connection ∇ on a vector bundle E has the local exponents ν dzz m , . . . , ν r dzz m at p if and only if there is an endomorphism N ∈ End( E | mp ) whose eigenvalues are µ , . . . , µ r and ν ( N ) dzz m = ∇| mp .For the construction of the unfolded moduli space of connections, we introduce a notion of ( ν , µ )-connection. Let C be a complex smooth projective curve of genus g and D = D (1) (cid:116) · · · (cid:116) D ( n ) be adivisor on C locally given by the equation D ( i ) = { z m i − (cid:15) m i = 0 } . The local exponents ν = ( ν ( i ) ( T ))and µ = ( µ ( i ) k ) are given by ν ( i ) ( T ) ∈ O D ( i ) [ T ] and distinct complex numbers µ ( i )1 , . . . , µ ( i ) r ∈ C . Thedefinition of ( ν , µ )-connection is given in Definition 2.3 as a tuple ( E, ∇ , { N ( i ) } ), where E is an algebraicvector bundle on C , ∇ is a connection on E admitting poles along D and N ( i ) ∈ End( E | D ( i ) ) satisfies ∇| D ( i ) = ν ( N ( i ) ) dzz m i − (cid:15) m i and ϕ ( i ) µ ( N ( i ) ) = 0, where ϕ ( i ) µ ( T ) = ( T − µ ( i )1 ) · · · ( T − µ ( i ) r ). In subsection 5.1,we define the relative moduli space M α C , D (˜ ν , µ ) −→ T µ , λ of α -stable ( ν , µ )-connections, whose existence isprovided by Theorem 2.11. Here T µ , λ −→ ∆ (cid:15) is constructed in subsection 5.1, on which there are a fullfamily of pointed curves ( C, t , . . . , t n ), divisors D ( i ) given by the local equation z m i − (cid:15) m i = 0 and a fullfamily of exponents ν . The fiber of the moduli space M α C , D (˜ ν , µ ) over (cid:15) (cid:54) = 0 is a moduli space of regularsingular connections and the fiber over (cid:15) = 0 is a moduli space of generic unramified irregular singularconnections.The fiber M α C , D (˜ ν , µ ) (cid:15) =0 over (cid:15) = 0 ∈ ∆ (cid:15) is the moduli space of unramified irregular singular connections.In [19], we construct an algebraic splittingΨ : ( π T ν , λ ,(cid:15) =0 ) ∗ T T ν , λ ,(cid:15) =0 −→ T M α C , D (˜ ν , µ ) (cid:15) =0 of the surjection dπ T ν , λ ,(cid:15) =0 : T M α C , D (˜ ν , µ ) (cid:15) =0 −→ ( π T ν , λ ,(cid:15) =0 ) ∗ T T ν , λ ,(cid:15) =0 , where T T µ , λ ,(cid:15) =0 and T M α C , D (˜ ν , µ ) (cid:15) =0 arethe tangent bundles of T µ , λ ,(cid:15) =0 and M α C , D (˜ ν , µ ) (cid:15) =0 , respectively. The splitting Ψ is the irregular singulargeneralized isomonodromic deformation arising from the theory by Jimbo, Miwa and Ueno in [21]. Theidea of the construction of Ψ is to construct a horizontal lift of the universal relative connection, whichis a first order infinitesimal extension of the relative connection with an integrability condition. We noticehere that the complete description of the Jimbo-Miwa-Ueno equation in [21] says that the essential numberof independent variables of the unramified irregular singular generalized isomonodromic deformation is3 g − (cid:80) ni =1 ( r ( m i −
1) + 1).One of the reasons of the difficulty in the confluence problem will be that the number 3 g − D ofindependent variables of the regular singular isomonodromic deformation is much smaller than the number3 g − (cid:80) ni =1 ( r ( m i −
1) + 1) of independent variables of the irregular singular generalized isomonodromic
NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 3 deformation. Here we have deg D = (cid:80) ni =1 m i , because the divisors are connected by a flat family. In thispaper, we try to extend the splitting Ψ locally to the unfolded moduli space M α C , D (˜ ν , µ ) via regarding T µ , λ as the space of independent variables. The main theorem of this paper is the following: Theorem 0.1.
For a general point x ∈ M α C , D (˜ ν , µ ) (cid:15) =0 satisfying Assumption 5.7 in subsection 5.3, thereexist an analytic open neighborhood M ◦ ⊂ M α C , D (˜ ν , µ ) of x whose image in T µ , λ is denoted by T ◦ , blocksof local horizontal lifts (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) defined in Definition 5.8 and a holomorphic homomorphism Ψ : ( π T ◦ ) ∗ T hol T ◦ / ∆ (cid:15) −→ T holM ◦ / ∆ (cid:15) depending on (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) , which is a splitting of the canonical surjection of the tangent bundles T M ◦ / ∆ (cid:15) dπ T ◦ −−−→ ( π T ◦ ) ∗ T T ◦ / ∆ (cid:15) , such that the restriction Ψ (cid:12)(cid:12) M α C , D (˜ ν , µ ) (cid:15) =0 ∩ M ◦ of Ψ to the irregular singularlocus coincides with the irregular singular generalized isomonodromic deformation Ψ hol (cid:12)(cid:12) M α C , D (˜ ν , µ ) (cid:15) =0 ∩ M ◦ . The main idea of the construction of Ψ in Theorem 0.1 is to consider the restriction ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) | ∆ × M ◦ of the universal family of connections to a local holomorphic disk ∆ containing D ( i ) and to extend it to afamily of connections on P admitting regular singularity along ∞ . We extend this family of connections on P to a family of integrable connections ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j on P × Spec C [ h ] / ( h ) depending on the data (˜Ξ ( i ) l,j ( z ))adjusting the residue part at ∞ . We glue the local integrable connections ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:12)(cid:12) ∆ × M ◦ and obtaina global horizontal lift of ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) | C M ◦ , which induces an unfolding in Theorem 0.1. In our unfoldedgeneralized isomonodromic deformation determined by Ψ, the monodromy along a loop surrounding wholethe unfolding divisor D ( i ) is preserved constant, but the local monodromy around each regular singular pointin D ( i ) is not preserved constant, because the local exponents are not constant. So our unfolded generalizedisomonodromic deformation does not mean the usual regular singular isomonodromic deformation. Wenotice that the splitting Ψ in the theorem is not canonical because it is essentially determined by the blocksof local horizontal lifts (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) constructed in subsection 4.2, which depend on the data (˜Ξ ( i ) l,j ( z ))adjusting the residue part and also on a fundamental solution commuting with the monodromy around ∞ .So we cannot expect the splitting Ψ to be defined globally on M α C , D (˜ ν , µ ). Moreover, we cannot expect theintegrability of the subbundle im Ψ ⊂ T holM ◦ / ∆ (cid:15) .The author’s hope was to construct the unfolding Ψ via adopting the asymptotic arguments in theunfolding theory established by Hurtubise, Lambert and Rousseau in a series of papers [29], [30], [14], [15].Unfortunately we cannot achieve in such an easy way, because we do not know that the unfolded Stokesmatrices defined in [15] are constant for our generalized isomonodromic deformation Ψ. This is anotherreason why the splitting Ψ cannot be extended globally. At the present, the framework of this paperis tentative because the moduli space M α C , D (˜ ν , µ ) dose not seem to be enough for the description of theunfolded generalized isomonodromic deformation. The author’s hope is to find a good replacement of themoduli space which describes our splitting Ψ adequately.The organization of this paper is the following.In section 1, we introduce a factorization V κ −→ V ∨ θ −→ V of a given linear endomorphism f : V −→ V whose minimal polynomial is of degree dim V . This gives the correspondence in Proposition 1.1 andProposition 1.2 between the linear endomorphisms f : V −→ V whose minimal polynomial is of maximaldegree and the pairs [( θ, κ )] with θ, κ symmetric. Using this correspondence, we can give in Proposition 1.5a certain kind of expression of the Kirillov-Kostant symplectic form on a GL r ( C ) adjoint orbit.In section 2, we introduce the notion of ( ν , µ )-connection which involves both a regular singular con-nection and an unramified irregular singular connection. We give a construction of the moduli space of( ν , µ )-connections essentially using the construction method in [16]. From the idea in section 1, we can seethat a ( ν , µ ) connection corresponds to a tuple ( E, ∇ , { θ ( i ) , κ ( i ) } ). Doing the deformation theory for thistuple, we can get the smoothness of the moduli space and a symplectic form. These are summarized inTheorem 2.11.In section 3, we give an introduction to the unfolding theory constructed by Hurtubise, Lambert andRousseau by means of the restriction to a most easy case when the perturbation of the singularity is givenby the equation z m − (cid:15) m = 0. We need a consideration on the flows given by dz/dt = e √− θ ( z m − (cid:15) m ) in MICHI-AKI INABA
Proposition 3.1. One of the main tool in the unfolding theory is a fundamental solution given in Theorem3.2 which has an asymptotic property estimated by flows given in Proposition 3.1.In section 4, we consider a family of connections ∇ on a holomorphic disk ∆ = { z ∈ C | | z | < } admitting poles along { z m − (cid:15) m = 0 } . Under some generic assumption on ∇ , we give an extension of ∇ as a family of connections on O ⊕ r P with a regular singularity along ∞ , whose connection matrix isgiven by A ( z ) dz/ ( z m − (cid:15) m ). Using linear algebraic argument, we obtain an adjusting data ˜Ξ l,j ( z ) suchthat ˜Ξ l,j ( z ) dz/ ( z m − (cid:15) m ) has no residue at ∞ . Then we can get a family of integrable connections on P × Spec C [ h ] / ( h ) given by a connection matrix ( A ( z ) + ¯ h ˜Ξ l,j ( z )) dz/ ( z m − (cid:15) m ) + B ( z ) d ¯ h in Proposition4.11, where B ( z ) is a matrix of multivalued functions.In section 5, we give the setting of the relative moduli space of ( ν , µ )-connections whose generic fiberis a moduli space of regular singular connections and a special fiber is a moduli space of unramified ir-regular singular connections. On the irregular singular fiber, we can define the generalized isomonodromicdeformation Ψ , which is basically determined by the Jimbo-Miwa-Ueno theory and precisely given in [19].The integrability of the irregular singular generalized isomonodromic deformation on P is proved in [21],which is extended to ramified case in [8]. We give in Theorem 5.6 an alternative proof of its integrabilityinvolving the higher genus case from the uniqueness property of its formulation. Gluing the local integrableconnections constructed in section 4, we construct a global horizontal lift in Proposition 5.11, which givesa local analytic lift of the unramified irregular singular generalized isomonodromic deformation and obtainTheorem 0.1. 1. An observation from linear algebra on a GL r ( C ) adjoint orbit In this section, we give a small remark on an adjoint orbit of GL r ( C ) on gl r ( C ). From the idea of theobservation in this section, we will get in section 2 a convenient parametrization of the local exponents ofconnections. Furthermore, we will get a pertinent expression of the relative symplectic form on an unfoldedmoduli space of connections on smooth projective curves in section 2.1.1. Factorization of a linear endomorphism whose minimal polynomial is of maximal degree.
Let V be a vector space over C of dimension r and µ , . . . , µ r ∈ C be mutually distinct complex numbers.If we consider the subvariety C ( µ , . . . , µ r ) := { f : V −→ V : linear map with the eigenvalues µ , . . . , µ r } of the affine space Hom C ( V, V ), then C ( µ , . . . , µ r ) is isomorphic to the GL r ( C )-adjoint orbit of the diagonalmatrix µ · · · ... . . . ... · · · µ r . So C ( µ , . . . , µ r ) has a symplectic structure given by the Kirillov-Kostant symplectic form. Indeed there isa canonical morphism from C ( µ , . . . , µ r ) to the complete flag variety F ( V ) by sending each f to the flagof V induced by the eigen space decomposition of f . The fiber is isomorphic to the set of upper triangularnilpotent matrices which is also isomorphic to the cotangent space of F ( V ). So C ( µ , . . . , µ r ) is locallyisomorphic over F ( V ) to the cotangent bundle over F ( V ) and the symplectic structure from the cotangentbundle coincides with the Kirillov-Kostant symplectic form. In subsection 1.2, we give another expressionof the symplectic form on the adjoint orbit C ( µ , . . . , µ r ). For the construction of the symplectic form, weextend to a slightly more general setting.Let ϕ ( T ) ∈ C [ T ] be a monic polynomial of degree r and V be a vector space over C of dimension r . Weput C ϕ ( T ) := { f : V −→ V | f is a linear map whose minimal polynomial is ϕ ( T ) } . Recall that ϕ ( T ) is a minimal polynomial of f : V −→ V if and only if ϕ ( f ) = 0 and the induced map C [ T ] / ( ϕ ( T )) (cid:51) P ( T ) (cid:55)→ P ( f ) ∈ End( V )is injective. Proposition 1.1.
For each f ∈ C ϕ ( T ) , there are an isomorphism θ : V ∨ ∼ −→ V and a linear map κ : V −→ V ∨ satisfying f = θ ◦ κ , t θ = θ and t κ = κ . Here V ∨ is the dual vector space of V , t θ : V ∨ −→ ( V ∨ ) ∨ = V is the dual of θ and t κ : V = ( V ∨ ) ∨ −→ V ∨ is the dual of κ . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 5
Proof.
The ring homomorphism C [ T ] (cid:51) P ( T ) (cid:55)→ P ( f ) ∈ End( V ) induces a C [ T ]-module structure on V .By an elementary theory of linear algebra, there is an isomorphism V ∼ −→ C [ T ] / ( ϕ ( T )) , of C [ T ]-modules, because the minimal polynomial ϕ ( T ) of f has degree r = dim V . Since the minimalpolynomial of t f coincides with ϕ ( T ), there is an isomorphism V ∨ ∼ −→ C [ T ] / ( ϕ ( T ))of C [ T ]-modules. So we can take an isomorphism θ : V ∨ ∼ −→ V of C [ T ]-modules. If we put κ := θ − ◦ f : V −→ V ∨ , then κ becomes a homomorphism of C [ T ]-modules and f = θ ◦ κ . We take a generator v ∗ ∈ V ∨ of V ∨ as a C [ T ]-module. Then v := θ ( v ∗ ) ∈ V is a generator of V as a C [ T ]-module. Take any w ∗ , w ∗ ∈ V ∨ . Then wecan write w ∗ = P ( t f ) v ∗ and w ∗ = P ( t f ) v ∗ for certain polynomials P ( T ) , P ( T ) ∈ C [ T ]. For the dualpairing (cid:104) , (cid:105) : V ∨ × V −→ C , we have (cid:104) w ∗ , t θ ( w ∗ ) (cid:105) = (cid:104) w ∗ ◦ θ, w ∗ (cid:105) = (cid:104) w ∗ , θ ( w ∗ ) (cid:105) = (cid:104) P ( t f ) v ∗ , θ ( P ( t f ) v ∗ ) (cid:105) = (cid:104) v ∗ ◦ P ( f ) , P ( f )( θ ( v ∗ )) (cid:105) = (cid:104) v ∗ , P ( f ) P ( f )( θ ( v ∗ )) (cid:105) = (cid:104) v ∗ , P ( f ) P ( f )( θ ( v ∗ )) (cid:105) = (cid:104) P ( t f ) v ∗ , θ ( P ( t f ) v ∗ ) (cid:105) = (cid:104) w ∗ , θ ( w ∗ ) (cid:105) . So we have t θ ( w ∗ ) = θ ( w ∗ ) and t θ = θ .Take any w , w ∈ V . Then there are polynomials P ( T ) , P ( T ) ∈ C [ T ] satisfying w = P ( f ) v and w = P ( f ) v . We have (cid:104) t κ ( w ) , w (cid:105) = (cid:104) κ ( w ) , w (cid:105) = (cid:104) κ ( P ( f ) v ) , P ( f ) v (cid:105) = (cid:104) θ − f P ( f ) v, P ( f ) v (cid:105) = (cid:104) t ( f P ( f )) θ − ( v ) , P ( f ) v (cid:105) = (cid:104) θ − ( v ) , f P ( f ) P ( f ) v (cid:105) = (cid:104) θ − ( v ) , f P ( f ) P ( f ) v (cid:105) = (cid:104) κ ( P ( f ) v ) , P ( f ) v (cid:105) = (cid:104) κ ( w ) , w (cid:105) . So we have t κ ( w ) = κ ( w ) and t κ = κ holds. (cid:3) Proposition 1.2.
For f ∈ C ϕ ( T ) , assume that θ , θ : V ∨ ∼ −→ V are isomorphisms and κ , κ : V −→ V ∨ are linear maps satisfying f = θ ◦ κ = θ ◦ κ , t θ = θ , t θ = θ , t κ = κ and t κ = κ . Then thereexists P ( T ) ∈ ( C [ T ] / ( ϕ ( T ))) × satisfying θ = θ ◦ P ( t f ) and κ = ( P ( t f )) − ◦ κ .Proof. Put σ := θ − ◦ θ : V ∨ −→ V ∨ . Then t f ◦ σ = t κ ◦ t θ ◦ θ − ◦ θ = κ ◦ θ ◦ θ − ◦ θ = κ ◦ θ and σ ◦ t f = θ − ◦ θ ◦ t κ ◦ t θ = θ − ◦ θ ◦ κ ◦ θ = θ − ◦ f ◦ θ = θ − ◦ θ ◦ κ ◦ θ = κ ◦ θ . So σ ◦ t f = t f ◦ σ and σ : V ∨ ∼ −→ V ∨ becomes a C [ T ]-isomorphism. Since C [ T ] / ( ϕ ( T )) ∼ −→ Hom C [ T ] ( V ∨ , V ∨ ),there exists P ( T ) ∈ ( C [ T ] / ( ϕ ( T ))) × satisfying P ( t f ) = σ = θ − ◦ θ . So we have θ ◦ P ( t f ) = θ , κ = θ − ◦ f = θ − ◦ θ ◦ κ = σ ◦ κ and κ = σ − ◦ κ = P ( t f ) − ◦ κ . (cid:3) An expression of the symplectic form on a GL r ( C ) adjoint orbit. Let the notations V , ϕ ( T ), r and C ϕ ( T ) be as in subsection 1.1. We set S ( V ∨ , V ) = (cid:8) θ ∈ Hom C ( V ∨ , V ) (cid:12)(cid:12) t θ = θ (cid:9) S ( V, V ∨ ) = (cid:8) κ ∈ Hom C ( V, V ∨ ) (cid:12)(cid:12) t κ = κ (cid:9) and S := (cid:26) ( θ, κ ) ∈ S ( V ∨ , V ) × S ( V, V ∨ ) (cid:12)(cid:12)(cid:12)(cid:12) θ is isomorphic, ϕ ( θ ◦ κ ) = 0 and the induced map C [ T ] / ( ϕ ( T )) (cid:51) P ( T ) (cid:55)→ P ( θ ◦ κ ) ∈ End( V ) is injective (cid:27) . MICHI-AKI INABA
Then there is an action of the commutative algebraic group ( C [ T ] / ( ϕ ( T ))) × on S defined by P ( T ) · ( θ, κ ) = ( θ ◦ P ( κ ◦ θ ) , P ( κ ◦ θ ) − ◦ κ ) . for P ( T ) ∈ ( C [ T ] / ( ϕ ( T ))) × . We can see by Proposition 1.1 and Proposition 1.2 that the quotient of S bythe action of ( C [ T ] / ( ϕ ( T ))) × is isomorphic to C ϕ ( T ) : S / ( C [ T ] / ( ϕ ( T ))) × ∼ = C ϕ ( T ) . We describe the tangent space of C ϕ ( T ) at f = θ ◦ κ via this isomorphism. Let us consider the complex(1) C [ T ] / ( ϕ ( T )) d −→ S ( V ∨ , V ) ⊕ S ( V, V ∨ ) d −→ ( C [ T ] / ( ϕ ( T ))) ∨ defined by d ( P ( T )) = ( θ ◦ P ( t f ) , − P ( t f ) ◦ κ ) (cid:0) P ( T ) ∈ C [ T ] / ( ϕ ( T )) (cid:1) d ( τ, ξ ) : C [ T ] / ( ϕ ( T )) (cid:51) P ( T ) (cid:55)→ Tr( P ( f ) ◦ ( θ ◦ ξ + τ ◦ κ )) ∈ C (cid:0) ( τ, ξ ) ∈ S ( V ∨ , V ) ⊕ S ( V, V ∨ ) (cid:1) . Proposition 1.3.
The tangent space T S ( θ, κ ) of S at ( θ, κ ) is isomorphic to ker d . Before proving the proposition, we prove the following lemma.
Lemma 1.4.
For f ∈ C ϕ ( T ) , the sequence −→ C [ T ] / ( ϕ ( T )) ι f −→ End C ( V ) ad( f ) −−−→ End C ( V ) π f −−→ ( C [ T ] / ( ϕ ( T ))) ∨ −→ is exact, where ι f is defined by ι f ( P ( T )) = P ( f ) and π f is the dual of ι f .Proof of Lemma 1.4. The map ι f is injective since f belongs to C ϕ ( T ) . Since the minimal polynomial of f is of degree r = dim V , the linear mapad( f ) : End C ( V ) (cid:51) g (cid:55)→ f ◦ g − g ◦ f ∈ End C ( V )satisfies ker ad( f ) = C [ f ] = im ι f . In particular, we have rank ad( f ) = r − r . The map π f is given by π f ( g )( P ( T )) = Tr( g ◦ P ( f ))for g ∈ End C ( V ) and P ( T ) ∈ C [ T ] / ( ϕ ( T )). So we have π f (ad( f )( g ))( P ( T )) = Tr(( f ◦ g − g ◦ f )( P ( f ))= Tr( P ( f ) ◦ f ◦ g ) − Tr( g ◦ f ◦ P ( f )) = Tr( f ◦ P ( f ) ◦ g ) − Tr( f ◦ P ( f ) ◦ g ) = 0for g ∈ End C ( V ) and P ( T ) ∈ C [ T ] / ( ϕ ( T )), which means π f ◦ ad( f ) = 0. So we haveim ad( f ) = ker π f = (cid:8) g ∈ End C ( V ) (cid:12)(cid:12) Tr( f i ◦ g ) = 0 for i = 0 , , . . . , r − (cid:9) , because the right hand side is of dimension r − r . Thus we have proved the lemma. (cid:3) Proof of Proposition 1.3.
If we take ( τ, ξ ) ∈ ker d , we have π f ( θ ◦ ξ + τ ◦ κ ) = d ( τ, ξ ) = 0. By Lemma 1.4,there is g ∈ End( V ) satisfying θ ◦ ξ + τ ◦ κ = f ◦ g − g ◦ f . We write ϕ ( T ) = b r T r + b r − T r − + · · · + b T + b with b r = 1. Then the C [ t ] / ( t )-valued point ( θ + τ ¯ t, κ + ξ ¯ t ) of S ( V ∨ , V ) × S ( V, V ∨ ) satisfies ϕ (( θ + τ ¯ t ) ◦ ( κ + ξ ¯ t )) = ϕ ( f + ( θ ◦ ξ + τ ◦ κ )¯ t ) = ϕ ( f + ( f ◦ g − g ◦ f )¯ t ) = r (cid:88) i =0 b i ( f + ( f ◦ g − g ◦ f )¯ t ) i = r (cid:88) i =0 b i f i + i − (cid:88) j =0 f j ( f ◦ g − g ◦ f ) f i − j − ¯ t = r (cid:88) i =0 b i (cid:0) f i + ( f i ◦ g − g ◦ f i )¯ t (cid:1) = ϕ ( f ) + ( ϕ ( f ) ◦ g − g ◦ ϕ ( f ))¯ t = 0 . So ( θ + τ ¯ t, κ + ξ ¯ t ) gives a tangent vector of S at ( θ, κ ).Conversely take a tangent vector of S and let ( θ + τ ¯ t, κ + ξ ¯ t ) be the corresponding C [ t ] / ( t )-valued pointof S . Then we have ϕ (( θ + τ ¯ t ) ◦ ( κ + ξ ¯ t )) = 0 and(2) C [ t ] / ( t )[ T ] / ( ϕ ( T )) (cid:51) P ( T ) (cid:55)→ P (( θ + τ ¯ t ) ◦ ( κ + ξ ¯ t )) ∈ End C [ t ] / ( t ) ( V ⊗ C C [ t ] / ( t ))is injective, whose cokernel is flat over C [ t ] / ( t ). Recall that there is an isomorphism σ : C [ T ] / ( ϕ ( T )) ∼ −→ V .So we can take a generator v = σ (1) of V as a C [ T ]-module. If we take a lift ˜ v ∈ V ⊗ C [ t ] / ( t ) of v , then NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 7 ˜ v becomes a generator of V ⊗ C [ t ] / ( t ) as a C [ t ] / ( t )[ T ]-module with respect to the action of C [ t ] / ( t )[ T ]induced by the ring homomorphism (2). So we have an isomorphism˜ σ : C [ t ] / ( t )[ T ] / ( ϕ ( T )) ∼ −→ V ⊗ C [ t ] / ( t )satisfying ˜ σ (1) = ˜ v . If we denote by id the identity map, σ ⊗ id : C [ T ] / ( ϕ ( T )) ⊗ C [ t ] / ( t ) ∼ −→ V ⊗ C C [ t ] / ( t )is another C [ t ] / ( t )[ T ]-isomorphism with respect to the action of C [ t ] / ( t )[ T ] on V ⊗ C C [ t ] / ( t ) via the ringhomomorphism C [ t ] / ( t )[ T ] (cid:51) P ( T ) (cid:55)→ P ( θ ◦ κ ⊗ id) ∈ End C [ t ] / ( t ) ( V ⊗ C [ t ] / ( t )) . Composing ˜ σ − with σ ⊗ id, we obtain a C [ t ] / ( t )-automorphism of V ⊗ C [ t ] / ( t ) of the form id + Q ¯ t with Q ∈ End C ( V ) which makes the diagram V ⊗ C C [ t ] / ( t ) ( θ + τ ¯ t ) ◦ ( κ + ξ ¯ t ) −−−−−−−−−→ V ⊗ C C [ t ] / ( t ) id+ Q ¯ t (cid:121) id+ Q ¯ t (cid:121) V ⊗ C C [ t ] / ( t ) θ ◦ κ ⊗ id −−−−−→ V ⊗ C C [ t ] / ( t )commutative. Then we have( θ ◦ ξ + τ ◦ κ )¯ t = ( θ + τ ¯ t ) ◦ ( κ + ξ ¯ t ) − θ ◦ κ = (id − Q ¯ t ) ◦ ( θ ◦ κ ) ◦ (id + Q ¯ t ) − θ ◦ κ = ( f ◦ Q − Q ◦ f )¯ t and Tr( f i ◦ ( θ ◦ ξ + τ ◦ κ )) = Tr( f i ( f ◦ Q − Q ◦ f )) = Tr( f i +1 ◦ Q − Q ◦ f i +1 ) = 0for any i ≥
0. Thus we have ( τ, ξ ) ∈ ker d . By the correspondence ( τ, ξ ) (cid:55)→ ( θ + τ ¯ t, κ + ξ ¯ t ), we get theisomorphism from ker d to the tangent space of S at ( θ, κ ). (cid:3) We can see that im( d ) coincides with the tangent space of the ( C [ T ] / ( ϕ ( T )) × -orbit of ( θ, κ ) in S . Sothe tangent space of C ϕ ( T ) = S / ( C [ T ] / ( ϕ ( T )) × at f = θ ◦ κ is isomorphic to T S ( θ, κ ) / im d which is thefirst cohomology of the complex (1): T C ϕ ( T ) ( f ) ∼ = H (cid:18) C [ T ] / ( ϕ ( T )) d −→ S ( V ∨ , V ) ⊕ S ( V, V ∨ ) d −→ ( C [ T ] / ( ϕ ( T ))) ∨ (cid:19) . We define a pairing ω C ϕ ( T ) : T C ϕ ( T ) ( f ) × T C ϕ ( T ) ( f ) −→ C by(3) ω C ϕ ( T ) ([( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )]) = 12 Tr( τ ◦ ξ (cid:48) − τ (cid:48) ◦ ξ ) . If [( τ, ξ )] = 0, then we can write τ = θ ◦ P ( t f ) and ξ = − P ( t f ) ◦ κ . So we haveTr( τ ◦ ξ (cid:48) − τ (cid:48) ◦ ξ ) = Tr( θ ◦ P ( t f ) ◦ ξ (cid:48) + τ (cid:48) ◦ P ( t f ) ◦ κ ) = Tr( P ( f ) ◦ ( θ ◦ ξ (cid:48) + τ (cid:48) ◦ κ )) = 0 . Similarly we can see that Tr( τ ◦ ξ (cid:48) − τ (cid:48) ◦ ξ ) = 0 if [( τ (cid:48) , ξ (cid:48) )] = 0. Thus the pairing (3) is well-defined.On the other hand, there is a well-known symplectic form so called the Kirillov-Kostant form. For twotangent vectors [( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )] ∈ T C ϕ ( T ) ( f ) of C ϕ ( T ) at f = θ ◦ κ , we can see by Lemma 1.4 that thereexist g, g (cid:48) ∈ Hom(
V, V ) satisfying f ◦ g − g ◦ f = θ ◦ ξ + τ ◦ κ and f ◦ g (cid:48) − g (cid:48) ◦ f = θ ◦ ξ (cid:48) + τ (cid:48) ◦ κ . TheKirillov-Kostant symplectic form ω K-K is defined in [27, page 5, Definition 1] by ω K-K ([( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )]) = Tr( f ◦ ([ g, g (cid:48) ])) . Proposition 1.5.
The pairing ω C ϕ ( T ) defined in (3) coincides with the Kirillov-Kostant symplectic form ω K-K on the adjoint orbit C ϕ ( T ) .Proof. Take any member ( a, b ) ∈ S ( V ∨ , V ) ⊕ S ( V, V ∨ ) satisfying θ ◦ b + a ◦ κ = 0. Then we have( θ + a ¯ t ) ◦ ( κ + b ¯ t ) = θ ◦ κ = f ∈ End C [ t ] / ( t ) ( V ⊗ C C [ t ] / ( t )) , from which we can see( κ + b ¯ t ) ◦ ( θ + a ¯ t ) = t ( κ + b ¯ t ) ◦ t ( θ + a ¯ t ) = t (( θ + a ¯ t ) ◦ ( κ + b ¯ t )) = t f = κ ◦ θ. So we have(id + θ − a ¯ t ) ◦ t f = θ − ◦ ( θ + a ¯ t ) ◦ ( κ + b ¯ t ) ◦ ( θ + a ¯ t )= θ − ◦ θ ◦ κ ◦ ( θ + a ¯ t ) = κ ◦ θ + κ ◦ θ ◦ θ − ◦ a ¯ t = t f ◦ (id + θ − ◦ a ¯ t ) . MICHI-AKI INABA
Then we have θ − ◦ a ∈ End C [ T ] ( V ∨ ) ∼ = C [ T ] / ( ϕ ( T )) and there exists P ( T ) ∈ C [ T ] / ( ϕ ( T )) satisfying θ − a = P ( t f ). So we have a = θ ◦ P ( t f ) and b = − θ − ◦ a ◦ κ = − P ( t f ) ◦ κ , which mean that( a, b ) ∈ im( d ). Thus we have proved(4) im( d ) = ker (cid:16) S ( V ∨ , V ) ⊕ S ( V, V ∨ ) (cid:51) ( a, b ) (cid:55)→ θ ◦ b + a ◦ κ ∈ Hom(
V, V ) (cid:17) . Take two tangent vectors [( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )] ∈ T C ϕ ( T ) ( f ) of C ϕ ( T ) at f = θ ◦ κ . Since ( τ, ξ ) , ( τ (cid:48) , ξ (cid:48) ) ∈ ker d ,we can see from Lemma 1.4 that there exist g, g (cid:48) ∈ Hom(
V, V ) satisfying f ◦ g − g ◦ f = θ ◦ ξ + τ ◦ κ and f ◦ g (cid:48) − g (cid:48) ◦ f = θ ◦ ξ (cid:48) + τ (cid:48) ◦ κ . Note that we have θ ◦ ( κ ◦ g (cid:48) + t g (cid:48) ◦ κ ) + ( − g (cid:48) ◦ θ − θ ◦ t g (cid:48) ) ◦ κ = θ ◦ κ ◦ g (cid:48) − g (cid:48) ◦ θ ◦ κ = f ◦ g (cid:48) − g (cid:48) ◦ f = θ ◦ ξ (cid:48) + τ (cid:48) ◦ κ. By the equality (4), we have [( τ (cid:48) , ξ (cid:48) )] = [( − g (cid:48) ◦ θ − θ ◦ t g (cid:48) , κ ◦ g (cid:48) + t g (cid:48) ◦ κ )] in T C ϕ ( T ) ( f ) and we may assumethat τ (cid:48) = − g (cid:48) ◦ θ − θ ◦ t g (cid:48) and ξ (cid:48) = κ ◦ g (cid:48) + t g (cid:48) ◦ κ . We have ω K-K ([( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )]) = Tr( f ◦ ([ g, g (cid:48) ])) = Tr( f ◦ ( g ◦ g (cid:48) − g (cid:48) ◦ g ))= Tr(( f ◦ g − g ◦ f ) ◦ g (cid:48) + ( g ◦ f ◦ g (cid:48) − f ◦ g (cid:48) ◦ g ))= Tr(( θ ◦ ξ + τ ◦ κ ) ◦ g (cid:48) ) + Tr( g ◦ ( f ◦ g (cid:48) ) − ( f ◦ g (cid:48) ) ◦ g )= Tr(( θ ◦ ξ + τ ◦ κ ) ◦ g (cid:48) ) = Tr( g (cid:48) ◦ θ ◦ ξ ) + Tr( τ ◦ κ ◦ g (cid:48) )= 12 (cid:0) Tr( g (cid:48) ◦ θ ◦ ξ ) + Tr( t ξ ◦ t θ ◦ t g (cid:48) ) + Tr( τ ◦ κ ◦ g (cid:48) ) + Tr( t g (cid:48) ◦ t κ ◦ t τ ) (cid:1) . Claim 1.6.
Tr( u ◦ v ) = Tr( v ◦ u ) for any u ∈ Hom(
V, V ∨ ) and any v ∈ Hom( V ∨ , V ) . Using the above claim, we have Tr( t ξ ◦ t θ ◦ t g (cid:48) ) = Tr( t θ ◦ t g (cid:48) ◦ t ξ ) = Tr( θ ◦ t g (cid:48) ◦ ξ ) and Tr( t g (cid:48) ◦ t κ ◦ t τ )) =Tr( t τ ◦ t g (cid:48) ◦ t κ ) = Tr( τ ◦ t g (cid:48) ◦ κ ). So we have ω K-K ([( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )]) = 12 (cid:0) Tr( g (cid:48) ◦ θ ◦ ξ ) + Tr( t ξ ◦ t θ ◦ t g (cid:48) ) + Tr( τ ◦ κ ◦ g (cid:48) ) + Tr( t g (cid:48) ◦ t κ ◦ t τ ) (cid:1) = 12 (cid:0) Tr(( g (cid:48) ◦ θ + θ ◦ t g (cid:48) ) ◦ ξ ) + Tr( τ ◦ ( κ ◦ g (cid:48) + t g (cid:48) ◦ κ )) (cid:1) = 12 (Tr( − τ (cid:48) ◦ ξ ) + Tr( τ ◦ ξ (cid:48) )) = ω C ϕ ( T ) ([( τ, ξ )] , [( τ (cid:48) , ξ (cid:48) )]) . For the proof of Claim 1.6, we take a basis e , . . . , e r of V and its dual basis e ∗ , . . . , e ∗ r of V ∨ . If write u ( e j ) = (cid:80) ri =1 a ij e ∗ i and v ( e ∗ l ) = (cid:80) rk =1 b kl e k , then we haveTr( u ◦ v ) = Tr r (cid:88) i,l =1 r (cid:88) k =1 a ik b kl e ∗ i ⊗ e l = r (cid:88) k =1 r (cid:88) i =1 a ik b ki Tr( v ◦ u ) = Tr r (cid:88) j,k =1 r (cid:88) i =1 a ij b ki e k ⊗ e ∗ j = r (cid:88) i =1 r (cid:88) k =1 a ik b ki So we have Tr( u ◦ v ) = Tr( v ◦ u ) and Claim 1.6 follows. Thus we have proved ω K-K = ω C ϕ ( T ) . (cid:3) Algebraic construction of an unfolding of the moduli space of unramified irregularsingular connections
Regular singular and unramified irregular singular connections as ( ν , µ ) -connections. Let C be a complex smooth projective irreducible curve of genus g . We take an effective divisor D ⊂ C , whichhas a decomposition D = D (1) + D (2) + · · · + D ( n ) = D (1) (cid:116) · · · (cid:116) D ( n ) , where each D ( i ) is an effective divisorof degree m i and D ( i ) ∩ D ( i (cid:48) ) = ∅ for i (cid:54) = i (cid:48) . We write D ( i ) = p ( i )1 + p ( i )2 + · · · + p ( i ) m i for 1 ≤ i ≤ n , whereeach p ( i ) j is a reduced point in C and it may be possible that p ( i ) j = p ( i ) j (cid:48) for j (cid:54) = j (cid:48) .Using the Chinese remainder theorem O D ( i ) ∼ = (cid:89) p ∈ D ( i ) O D ( i ) ,p , we can choose ¯ z ( i ) ∈ O D ( i ) satisfying ¯ z ( i ) ( p ( i ) j ) (cid:54) = ¯ z ( i ) ( p ( i ) j (cid:48) ) for p ( i ) j (cid:54) = p ( i ) j (cid:48) and d ¯ z ( i ) | p ( i ) j (cid:54) = 0 ∈ Ω C | p ( i ) j for j = 1 , . . . , m i . We write ¯ z ( i ) j := ¯ z ( i ) − ¯ z ( i ) ( p ( i ) j ), where ¯ z ( i ) ( p ( i ) j ) ∈ C is the value of ¯ z ( i ) at p ( i ) j . We take NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 9 local lifts z ( i ) ∈ O C of ¯ z ( i ) , put z ( i ) j := z ( i ) − z ( i ) ( p ( i ) j ) and define(5) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i := dz ( i ) z ( i )1 z ( i )2 · · · z ( i ) m i (cid:12)(cid:12)(cid:12)(cid:12) D ( i ) ∈ Ω C ( D ) | D ( i ) which becomes a local basis of Ω C ( D ) | D ( i ) . Note that the above definition is independent of the choice ofrepresentatives z ( i ) of ¯ z ( i ) . We denote the multiplicity of D ( i ) at each p ∈ D ( i ) by m ( i ) p . If l , . . . , l m i areintegers satisfying 0 ≤ l , . . . , l m i ≤
1, there is a unique decomposition(6) d ¯ z ( i ) (¯ z ( i )1 ) l (¯ z ( i )2 ) l · · · (¯ z ( i ) m i ) l mi = (cid:88) p ∈ D ( i ) (cid:88) ≤ j ≤ m ( i ) p a ( i ) p,j d ¯ z ( i ) (¯ z ( i ) − ¯ z ( i ) ( p ))) j with a ( i ) p,j ∈ C . Since a ( i ) p,j is determined by a ( i ) p,j = lim z ( i ) → p m ( i ) p − j )! d m ( i ) p − j d ( z ( i ) ) m ( i ) p − j (cid:32) ( z ( i ) − z ( i ) ( p )) m ( i ) p z ( i )1 · · · z ( i ) m i (cid:33) , we can see that a ( i ) p,j is independent of the choice of the lift z ( i ) of ¯ z ( i ) . Then we define res p (cid:32) d ¯ z ( i ) (¯ z ( i )1 ) l · · · (¯ z ( i ) m i ) l mi (cid:33) := a ( i ) p, . Lemma 2.1. If l , . . . , l m i are integers satisfying ≤ l , . . . , l m i ≤ and l + · · · + l m i ≥ , the equality (cid:88) p ∈ D ( i ) res p (cid:32) d ¯ z ( i ) (¯ z ( i )1 ) l · · · (¯ z ( i ) m i ) l mi (cid:33) = 0 holds.Proof. It is sufficient to prove the equality for the case l = l = · · · = l m i = 1. Since the equality whichwe want is a formal equality determined by (6), it is sufficient to prove the equality(7) (cid:88) p ∈{ p ,...,p m } res z = p (cid:18) dz ( z − p )( z − p ) · · · ( z − p m ) (cid:19) = 0when z is a coordinate of the complex plane C , m ≥ p . . . , p m ∈ C may not be distinct. If we take acircle γ in C which is a boundary of a large disk containing all the points p , . . . , p m within, then we have (cid:88) p ∈{ p ,...,p m } res z = p (cid:18) dz ( z − p )( z − p ) · · · ( z − p m ) (cid:19) = 12 π √− (cid:90) γ dz ( z − p )( z − p ) · · · ( z − p m )= − res z = ∞ (cid:18) dz ( z − p )( z − p ) · · · ( z − p m ) (cid:19) = 0because m ≥
2. Thus the equality (7) holds. (cid:3)
We take µ = ( µ ( i ) j ) ≤ i ≤ n ≤ j ≤ r ∈ H ( D ( i ) , O nrD ( i ) ) such that µ ( i )1 | p , . . . , µ ( i ) r | p are mutually distinct at any point p ∈ D ( i ) . Then we define a polynomial ϕ ( i ) µ ( T ) ∈ H ( D ( i ) , O D ( i ) )[ T ] by setting ϕ ( i ) µ ( T ) := r (cid:89) k =1 ( T − µ ( i ) k ) . We fix a tuple of complex numbers λ = ( λ ( i ) k ) ≤ i ≤ n ≤ k ≤ r ∈ C nr satisfying (cid:80) ni =1 (cid:80) rk =1 λ ( i ) k ∈ Z and put a := − n (cid:88) i =1 r (cid:88) k =1 λ ( i ) k . For each i , we take a polynomial ν ( i ) ( T ) = c ( i )0 + c ( i )1 T + · · · + c ( i ) r − T r − ∈ H ( D ( i ) , O D ( i ) )[ T ] such that theexpression ν ( i ) ( µ ( i ) k ) = (cid:88) ≤ l ,...,lmi ≤ , ≤ l ··· + lmi For each i , ν ( i ) ( µ ( i )1 ) (cid:12)(cid:12) p , . . . , ν ( i ) ( µ ( i ) r ) (cid:12)(cid:12) p are mutually distinct at any point p ∈ D ( i ) . Definition 2.3. We say that a tuple ( E, ∇ , { N ( i ) } ≤ i ≤ n ) is a ( ν , µ )-connection on ( C, D ) if(i) E is an algebraic vector bundle on C of rank r and degree a ,(ii) ∇ : E −→ E ⊗ Ω C ( D ) is an algebraic connection on E admitting poles along D ,(iii) N ( i ) : E | D ( i ) −→ E | D ( i ) is an O D ( i ) -homomorphism satisfying ϕ ( i ) µ ( N ( i ) ) = 0, the homomorphism(10) O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) ∈ End( E | D ( i ) )is injective and ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) for 1 ≤ i ≤ n . Remark 2.4. The injectivity of the homomorphism (10) in Definition 2.3 implies that O D ( i ) [ T ] / ( ϕ ( i ) µ ( T ))becomes an O D ( i ) -subbundle of End( E | D ( i ) ). Proposition 2.5. Assume that D is a reduced divisor on C . In other words, we assume that p ( i ) j (cid:54) = p ( i ) j (cid:48) for j (cid:54) = j (cid:48) . Then giving a ( ν , µ ) -connection on ( C, D ) is equivalent to giving a regular singular connection ( E, ∇ ) on C admitting poles along D whose residue res p ( i ) j ( ∇ ) at p ( i ) j has the distinct eigenvalues (cid:40) ν ( i ) ( µ ( i ) k ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ r (cid:41) . Proof. Let ( E, ∇ , { N ( i ) } ) be a ( ν , µ )-connection on ( C, D ). The restriction N ( i ) | p ( i ) j : E | p ( i ) j −→ E | p ( i ) j of N ( i ) to the fiber E | p ( i ) j of E at p ( i ) j satisfies (cid:81) rk =1 ( N ( i ) | p ( i ) j − µ ( i ) k id E | p ( i ) j ) = 0, because ϕ ( i ) µ ( N ( i ) ) = 0. Fromthe injectivity of the homomorphism (10) in Definition 2.3, the induced homomorphism C [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) | p ( i ) j ) ∈ End( E | p ( i ) j )is injective. So N ( i ) | p ( i ) j has the distinct eigenvalues µ ( i )1 | p ( i ) j , . . . , µ ( i ) r | p ( i ) j . By Assumption 2.2, the linearendomorphism on E | p ( i ) j ν ( i ) ( N ( i ) ) | p ( i ) j = c ( i )0 | p ( i ) j id E | p ( i ) j + c ( i )1 | p ( i ) j N ( i ) | p ( i ) j + · · · + c ( i ) r | p ( i ) j ( N ( i ) | p ( i ) j ) m i r − r : E | p ( i ) j −→ E | p ( i ) j NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 11 has the distinct eigenvalues ν ( i ) ( µ ( i )1 ) | p ( i ) j , . . . , ν ( i ) ( µ ( i ) r ) | p ( i ) j . Since ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) , theresidue homomorphism res p ( i ) j ( ∇ ) : E | p ( i ) j −→ E | p ( i ) j has the eigenvalues (cid:40) ν ( i ) ( µ ( i ) k ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ r (cid:41) . Conversely let E be a vector bundle on C of rank r and ∇ : E −→ E ⊗ Ω C ( D ) be a connection whoseresidue res p ( i ) j ( ∇ ) at p ( i ) j has the distinct eigenvalues (cid:40) ν ( i ) ( µ ( i ) k ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ r (cid:41) .Since the diagonal matrix R = ν ( i ) ( µ ( i )1 ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) · · · ... . . . ... · · · ν ( i ) ( µ ( i ) r ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) has the distinct eigenvalues and commutes with the diagonal matrix N = µ ( i )1 | p ( i ) j · · · ... . . . ... · · · µ ( i ) r | p ( i ) j , thematrix N can be written as a polynomial ψ ( i ) j ( R ) in R with coefficients in C , that is, N = ψ ( i ) j ( R ). Considerthe linear map ψ ( i ) j ( res p ( i ) j ( ∇ )) : E | p ( i ) j −→ E | p ( i ) j . By the Chinese remainder theorem O D ( i ) ∼ −→ (cid:76) m i j =1 O p ( i ) j , we have an isomorphismHom O D ( i ) ( E | D ( i ) , E | D ( i ) ) ∼ −→ m i (cid:77) j =1 Hom O p ( i ) j ( E | p ( i ) j , E | p ( i ) j ) . So there is an endomorphism N ( i ) : E | D ( i ) −→ E | D ( i ) satisfying N ( i ) | p ( i ) j = ψ ( i ) j ( res p ( i ) j ( ∇ )) for 1 ≤ j ≤ m i .Since R = ν ( i ) ( N ) | p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) = ν ( i ) ( ψ ( i ) j ( R )) res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) , we can see res p ( i ) j ( ∇ ) = ν ( i ) (cid:16) ψ ( i ) j ( res p ( i ) j ( ∇ ) (cid:17) res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) = ν ( i ) ( N ( i ) ) (cid:12)(cid:12) p ( i ) j res p ( i ) j (cid:32) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:33) for 1 ≤ j ≤ m i , which is equivalent to ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) . From the definition, each N ( i ) | p ( i ) j has the distinct eigenvalues µ ( i )1 | p ( i ) j , . . . , µ ( i ) r | p ( i ) j and so the identity ϕ ( i ) µ ( N ( i ) ) = 0 follows. Thus( E, ∇ , { N ( i ) } ) becomes a ( ν , µ )-connection. (cid:3) The following definition of unramified irregular singular parabolic connection is given in [19]. Here werestrict to the case of generic exponents and a notation of suffix is slightly changed. Definition 2.6. Let t , . . . , t n ∈ C be distinct points and m , . . . , m n be integers satisfying m i > i . Take a generator z i ∈ m t i of the maximal ideal m t i of O C,t i . Assume that ν ( i )1 , . . . , ν ( i ) r ∈ O m i t i satisfy ν ( i ) k | t i (cid:54) = ν ( i ) k (cid:48) | t i for k (cid:54) = k (cid:48) . Then ( E, ∇ , { l ( i ) k } ) is said to be an unramified irregular singular parabolicconnection with the exponents ν ( i )1 dz i z m i i , . . . , ν ( i ) r dz i z m i i at t i if E is an algebraic vector bundle on C , ∇ : E −→ E ⊗ Ω C ( (cid:80) ni =1 m i t i ) is an algebraic connection, E | m i t i = l ( i )1 ⊃ l ( i )2 ⊃ · · · ⊃ l ( i ) r ⊃ l ( i ) r +1 = 0 is a filtrationsatisfying l ( i ) k /l ( i ) k +1 ∼ = O m i t i and (cid:16) ∇| m i t i − ν ( i ) k dz i z m i i id (cid:17) ( l ( i ) k ) ⊂ l ( i ) k +1 dz i z m i i for any k . Remark 2.7. Assume that ( E, ∇ , { l ( i ) k } ) is an unramified irregular singular parabolic connection with theexponents ν ( i )1 dz i z m i i , . . . , ν ( i ) r dz i z m i i in Definition 2.6 satisfying ν ( i ) k | t i (cid:54) = ν ( i ) k (cid:48) | t i for k (cid:54) = k (cid:48) . Then we can see asin the proof of [19, Proposition 2.3] that there is a decomposition(11) E | m i t i = r (cid:77) k =1 ker (cid:18) ∇| m i t i − ν ( i ) k dz i z m i i (cid:19) which induces the filtration l ( i ) ∗ and the diagonal representation matrix of ∇| m i t i ν ( i )1 dz i z m i i · · · ... . . . . . . · · · ν ( i ) r dz i z m i i with respect to a basis of E | m i t i obtained from the decomposition (11). Proposition 2.8. Under Assumption 2.2, suppose that each D ( i ) is a multiple divisor of degree m i for ≤ i ≤ n . In other words, we assume that p ( i ) j = p ( i ) j (cid:48) for any j, j (cid:48) and D ( i ) = m i p ( i )1 . Then giving a ( ν , µ ) -connection on ( C, D ) is equivalent to giving an unramified irregular singular parabolic connection ( E, ∇ , { l ( i ) k } ) on ( C, D ) with the exponents (cid:40) ν ( i ) ( µ ( i ) k ) d ¯ z ( i )1 (¯ z ( i )1 ) m i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ r (cid:41) at p ( i )1 .Proof. Assume that a ( ν , µ ) connection ( E, ∇ , { N ( i ) } ) on ( C, D ) is given. First note that there is a complex E | D ( i ) N ( i ) − µ ( i ) k −−−−−−→ E | D ( i ) (cid:81) k (cid:48)(cid:54) = k ( N ( i ) − µ ( i ) k (cid:48) ) −−−−−−−−−−−−→ E | D ( i ) which induces the homomorphism (cid:89) k (cid:48) (cid:54) = k ( N ( i ) − µ ( i ) k (cid:48) ) : coker( N ( i ) − µ ( i ) k ) −→ E | D ( i ) . By Remark 2.4, the restriction C [ T ] / ( ϕ µ ( T ) | p ( i )1 ) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) | p ( i )1 ) ∈ End( E | p ( i )1 ) of the homomor-phism (10) in Definition 2.3 to the reduced point p ( i )1 of D ( i ) = m i p ( i )1 is also injective. So N ( i ) | p ( i )1 : E | p ( i )1 −→ E | p ( i )1 has the distinct eigenvalues µ ( i )1 | p ( i )1 , . . . , µ ( i ) r | p ( i )1 and (cid:89) k (cid:48) (cid:54) = k ( N ( i ) − µ ( i ) k (cid:48) ) | p ( i )1 : coker(( N ( i ) − µ ( i ) k ) | p ( i )1 ) −→ E | p ( i )1 is an injection to the eigen subspace of E | p ( i )1 with respect to the eigenvalue µ ( i ) k | p ( i )1 of N ( i ) | p ( i )1 . Thereforewe can see that (cid:89) k (cid:48) (cid:54) = k ( N ( i ) − µ ( i ) k (cid:48) ) : coker( N ( i ) − µ ( i ) k ) −→ E | D ( i ) is also injective and its cokernel is a free O D ( i ) -module of rank r − 1. Socoker( N ( i ) − µ ( i ) k ) ∼ −→ ker( N ( i ) − µ ( i ) k ) ⊂ E | D ( i ) is a rank one subbundle of E | D ( i ) and we have a decomposition(12) E | D ( i ) = r (cid:77) k =1 ker( N ( i ) − µ ( i ) k ) . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 13 By the equality ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) , we can see that the representation matrix of ∇| D ( i ) withrespect to a basis giving the direct sum decomposition (12) of E | D ( i ) is ν ( i ) ( µ ( i )1 ) d ¯ z ( i ) (¯ z ( i )1 ) m i · · · ... . . . ... · · · ν ( i ) ( µ ( i ) r ) d ¯ z ( i ) (¯ z ( i )1 ) m i . If we choose the parabolic structure { l ( i ) k } compatible with the decomposition (12), then ( E, ∇ , { l ( i ) k } ) be-comes an unramified irregular singular parabolic connection with the exponents (cid:26) ν ( i ) ( µ ( i ) k ) d ¯ z ( i )1 (¯ z ( i )1 ) m i (cid:27) ≤ k ≤ r at p ( i )1 for 1 ≤ i ≤ n .Conversely, let ( E, ∇ , { l ( i ) k } ) be an unramified irregular singular parabolic connection with the exponents (cid:26) ν ( i ) ( µ ( i ) k ) d ¯ z ( i )1 (¯ z ( i )1 ) m i (cid:27) ≤ k ≤ r at p ( i )1 . Since ν ( i ) ( µ ( i )1 ) | p ( i )1 , . . . , ν ( i ) ( µ ( i ) r ) | p ( i )1 are mutually distinct, we have adecomposition E | D ( i ) = r (cid:77) k =1 ker (cid:32) ∇| D ( i ) − ν ( i ) ( µ ( i ) k ) d ¯ z ( i )1 (¯ z ( i )1 ) m i (cid:33) as in Remark 2.7 which is compatible with { l ( i ) k } . If we define a homomorphism N ( i ) : E | D ( i ) −→ E | D ( i ) bysetting N ( i ) (cid:12)(cid:12) ker (cid:0) ∇| D ( i ) − ν ( i ) ( µ ( i ) k ) d ¯ z ( i )( z ( i )1 ) mi (cid:1) = µ ( i ) k · id ker (cid:0) ∇| D ( i ) − ν ( i ) ( µ ( i ) k ) d ¯ z ( i )( z ( i )1 ) mi (cid:1) for each k , then N ( i ) satisfies ϕ ( i ) µ ( N ( i ) ) = 0 and ∇| D ( i ) = ν ( i ) ( N ( i ) ) d ¯ z ( i ) (¯ z ( i )1 ) m i . Since N ( i ) | p ( i )1 has the distincteigenvalues µ ( i )1 | p ( i )1 , . . . , µ ( i ) r | p ( i )1 , the homomorphism O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) ∈ End( E | D ( i ) )is injective, because of the injectivity of its restriction to the reduced point p ( i )1 of D ( i ) . So ( E, ∇ , { N ( i ) } )becomes a ( ν , µ )-connection. (cid:3) Now we come back to the general setting in Definition 2.3 and define a stability for a ( ν , µ )-connection( E, ∇ , { N ( i ) } ) which is necessary for the construction of the moduli space. By Assumption 2.2, there is aunique filtration(13) E | D ( i ) = l ( i )1 ⊃ l ( i )2 ⊃ · · · ⊃ l ( i ) r ⊃ l ( i ) r +1 = 0such that l ( i ) k /l ( i ) k +1 ∼ = O D ( i ) , (cid:16) ∇| D ( i ) − ν ( i ) ( µ ( i ) k ) dz i z m i i id (cid:17) ( l ( i ) k ) ⊂ l ( i ) k +1 dz i z m i i and ( N ( i ) − µ ( i ) k id)( l ( i ) k ) ⊂ l ( i ) k +1 forany i, k .We take a tuple of positive rational numbers α = ( α ( i ) k ) ≤ i ≤ n ≤ k ≤ r satisfying 0 < α ( i )1 < α ( i )2 < · · · < α ( i ) r < i and α ( i ) k (cid:54) = α ( i (cid:48) ) k (cid:48) for ( i, k ) (cid:54) = ( i (cid:48) , k (cid:48) ). The following definition in fact depends on the ordering of µ ( i )1 , . . . , µ ( i ) r . Definition 2.9. A ( ν , µ )-connection ( E, ∇ , { N ( i ) } ) on ( C, D ) is α -stable (resp. α -semistable) if the in-equalitydeg F + n (cid:88) i =1 r (cid:88) k =1 α ( i ) k length(( F | D ( i ) ∩ l ( i ) k ) / ( F | D ( i ) ∩ l ( i ) k +1 ))rank F < (resp. ≤ ) deg E + n (cid:88) i =1 r (cid:88) k =1 α ( i ) k length( l ( i ) k /l ( i ) k +1 )rank E holds for any subbundle 0 (cid:54) = F (cid:40) E satisfying ∇ ( F ) ⊂ F ⊗ Ω C ( D ), where { l ( i ) k } is the filtration (13) of E | D ( i ) determined by ∇| D ( i ) . Relative moduli space of (˜ ν , ˜ µ ) -connections. Let S be an irreducible algebraic variety over Spec C and let C −→ S be a smooth projective morphism whose geometric fibers are smooth projective irreduciblecurves of genus g . Assume that D is an effective Cartier divisor on C flat over S , which has a decomposition D = D (1) + · · · + D ( n ) = D (1) (cid:116) · · · (cid:116) D ( n ) , where D ( i ) is an effective Cartier divisor on C flat over S , which also has a decomposition D ( i ) = D ( i )1 + D ( i )2 + · · · + D ( i ) m i such that the composition D ( i ) j (cid:44) → C −→ S is isomorphic. Here we assume that D ( i ) ∩ D ( i (cid:48) ) = ∅ for i (cid:54) = i (cid:48) and ( D ( i ) j ) s ∩ ( D ( i ) j (cid:48) ) s = ∅ for j (cid:54) = j (cid:48) if ( D ( i ) j ) s , ( D ( i ) j (cid:48) ) s are generic fibers but D ( i ) j and D ( i ) j (cid:48) may intersect.Assume that we can take a section ¯ z ( i ) ∈ O D ( i ) such that ¯ z ( i ) − ¯ z ( i ) ( D ( i ) j ) = 0 is a defining equation of D ( i ) j in 2 D ( i ) and that d ¯ z ( i ) | p gives a local basis of Ω C /S ⊗ O D ( i ) | p for any point p ∈ D ( i ) , where ¯ z ( i ) ( D ( i ) j ) ∈ O S corresponds to ¯ z ( i ) | D ( i ) j via the isomorphism D ( i ) j ∼ −→ S . We denote ¯ z ( i ) − ¯ z ( i ) ( D ( i ) j ) ∈ O D ( i ) by ¯ z ( i ) j . Thenwe can define(14) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i ∈ Ω C /S ( D ( i ) ) | D ( i ) similarly to (5) which is a local basis of Ω C /S ( D ( i ) ) | D ( i ) .We fix ˜ µ = (˜ µ ( i ) j ) ≤ i ≤ n ≤ j ≤ r ∈ H ( D ( i ) , O nr D ( i ) ) such that ˜ µ ( i )1 | p , . . . , ˜ µ ( i ) r | p ∈ C are mutually distinct at anypoint p ∈ D ( i ) . Then we define a tuple ϕ ˜ µ = ( ϕ ( i )˜ µ ( T )) ≤ i ≤ n of polynomials by ϕ ( i )˜ µ ( T ) = r (cid:89) k =1 ( T − ˜ µ ( i ) k ) ∈ H ( D ( i ) , O D ( i ) )[ T ] . Assume that a ∈ Z and ˜ λ = (˜ λ ( i ) k ) ∈ H ( S, O S ) nr satisfying a + n (cid:88) i =1 r (cid:88) k =1 ˜ λ ( i ) k = 0are given. We also take a tuple ˜ ν = (˜ ν ( i ) ( T )) ≤ i ≤ n of polynomials˜ ν ( i ) ( T ) = c ( i )0 + c ( i )1 T + · · · + c ( i ) r − T r − ∈ H ( D ( i ) , O D ( i ) )[ T ]such that the expression ν ( i ) ( µ ( i ) k ) = (cid:88) ≤ l ,...,lmi ≤ ≤ l ··· + lmi We define a contravariant functor M α C , D (˜ ν , ˜ µ ) : (Sch /S ) o −→ (Sets) from the category(Sch /S ) of noetherian schemes over S to the category (Sets) of sets by setting M α C , D (˜ ν , ˜ µ )( S (cid:48) ) = (cid:110) ( E, ∇ , { N ( i ) } ≤ i ≤ n ) (cid:12)(cid:12)(cid:12) ( E, ∇ , { N ( i ) } ) satisfies the following (a),(b),(c),(d) (cid:111)(cid:46) ∼ , for a noetherian scheme S (cid:48) over S , where(a) E is a vector bundle on C S (cid:48) of rank r and deg( E | C s ) = a for any geometric point s of S ,(b) ∇ : E −→ E ⊗ Ω C S (cid:48) /S (cid:48) ( D S (cid:48) ) is an S (cid:48) -relative connection, in other words, ∇ ( f a ) = a ⊗ df + f ∇ ( a )for f ∈ O C S (cid:48) and a ∈ E , NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 15 (c) N ( i ) : E | D ( i ) S (cid:48) −→ E | D ( i ) S (cid:48) is an O D ( i ) S (cid:48) -homomorphim satisfying ϕ ( i ) µ ( N ( i ) ) = 0, the homomorphism O D ( i ) S (cid:48) [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) ∈ End( E | D ( i ) S (cid:48) )is an injection whose cokernel is flat over S (cid:48) , ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) S (cid:48) for 1 ≤ i ≤ n and(d) ( E | C s , ∇| C s , { N ( i ) | D ( i ) s } ) is α -stable for any geometric point s of S (cid:48) .Here ( E, ∇ , { N ( i ) } ) ∼ ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) if there are a line bundle L on S (cid:48) and an isomorphism σ : E ∼ −→ E (cid:48) ⊗L satisfying (id ⊗ σ ) ◦ ∇ = ∇ (cid:48) ◦ σ and σ | D ( i ) S (cid:48) ◦ N ( i ) = ( N (cid:48) ( i ) ⊗ id) ◦ σ | D ( i ) S (cid:48) for any i . Theorem 2.11. There exists a coarse moduli scheme M α C , D (˜ ν , ˜ µ ) of M α C , D (˜ ν , ˜ µ ) . The structure morphism M α C , D (˜ ν , ˜ µ ) −→ S is a smooth and quasi-projective morphism whose non-empty fiber is of dimension r ( g − 1) + 2 + r ( r − (cid:80) ni =1 m i . Moreover, there is a relative symplectic form on M α C , D (˜ ν , ˜ µ ) over S . We call M α C , D (˜ ν , ˜ µ ) in Theorem 2.11 the relative moduli space of α -stable (˜ ν , ˜ µ ) connections on ( C , D )over S . First we give a proof of the existence of the moduli space M α C , D (˜ ν , ˜ µ ). We define a moduli functor M : (Sch /S ) o −→ (Sets) by M ( S (cid:48) ) = (cid:110) ( E, ∇ , { l ( i ) k } ) (cid:12)(cid:12)(cid:12) ( E, ∇ , { l ( i ) k } ) satisfies the following (i),(ii),(iii),(iv) (cid:111) / ∼ for a noetherian scheme S (cid:48) over S , where(i) E is a vector bundle on C × S S (cid:48) of rank r and deg( E | C s ) = a for any geometric point s of S (cid:48) ,(ii) ∇ : E −→ E ⊗ Ω C S (cid:48) /S (cid:48) ( D S (cid:48) ) is a relative connection,(iii) E | D ( i ) S (cid:48) = l ( i )0 ⊃ l ( i )1 ⊃ · · · ⊃ l ( i ) r − ⊃ l ( i ) r = 0 is a filtration by coherent O D ( i ) S (cid:48) -submodules such thateach l ( i ) k /l ( i ) k +1 is flat over S (cid:48) and length(( l ( i ) k /l ( i ) k +1 ) | D ( i ) s ) = m i for any s ∈ S (cid:48) ,(iv) for any geometric point s of S (cid:48) , the fiber ( E, ∇ , { l ( i ) k } ) | C s satisfies the stability conditiondeg F + (cid:80) ni =1 (cid:80) rk =1 α ( i ) k length(( F | D ( i ) s ∩ l ( i ) k | D ( i ) s ) / ( F | D ( i ) s ∩ l ( i ) k +1 | D ( i ) s ))rank F< deg E | D ( i ) s + (cid:80) ni =1 (cid:80) rk =1 α ( i ) k length( l ( i ) k | D ( i ) s /l ( i ) k +1 | D ( i ) s )rank E for any subbundle 0 (cid:54) = F (cid:40) E | C s satisfying ∇| C s ( F ) ⊂ F ⊗ Ω C s ( D s ).Here ( E, ∇ , { l ( i ) k } ) ∼ ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) k } ) if there are a line bundle L on S (cid:48) and an isomorphism ( E, ∇ , { l ( i ) k } ) ∼ −→ ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) k } ) ⊗ O S (cid:48) L . Note that the parabolic structure { l ( i ) k } in (iii) has no relationship with the connec-tion ∇ in (ii). The following lemma is already used in [16], [17] and [19]. Lemma 2.12. There exists a coarse moduli scheme M of M . M is quasi-projective over S and representsthe ´etale sheafification of the moduli functor M .Proof. By [16, Theorem 5.1], there exists a relative coarse moduli scheme M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) over S ofparabolic Λ D -triples ( E , E , φ, ∇ , { l ( i ) k } ), where E and E are algebraic vector bundles of rank r on afiber of C over S , φ : E −→ E is an O C -homomorphism, ∇ : E −→ E ⊗ Ω C /S ( D ) satisfies ∇ ( f a ) = φ ( a ) ⊗ df + f ∇ ( a ) for f ∈ O C , a ∈ E , E | D ( i ) s = l ( i )0 ⊃ l ( i )1 ⊃ · · · ⊃ l ( i ) r = 0 is a filtration satisfyinglength( l ( i ) k /l ( i ) k +1 ) = m i and ( E , E , φ, ∇ , { l ( i ) k } ) satisfies a stability condition with respect to ( α (cid:48) , β , γ ).Furthermore, M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) is quasi-projective over S . The detail is written in [16, section 5]. Ifwe denote the moduli functor corresponding to M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) by M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) and choosean appropriate stability parameter ( α (cid:48) , β , γ ) by a similar argument to that in [16, section 5], then we candefine a morphism of functors M −→ M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) given by ( E, ∇ , { l ( i ) k } ) (cid:55)→ ( E, E, id E , ∇ , { l ( i ) k } ) which is represented by an open immersion. So there is aZariski open subset M ⊂ M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) satisfying M ∼ = M × M D , α (cid:48) , β ,γ C /S ( r,a, { m i } ) M D , α (cid:48) , β ,γ C /S ( r, a, { m i } ) . Then M represents the ´etale sheafification of M and becomes a coarse moduli scheme of M . (cid:3) Proof of the existence of M α C , D (˜ ν , ˜ µ ) . For some quasi-finite ´etale covering ˜ M −→ M , there is a universal family ( ˜ E, ˜ ∇ , { ˜ l ( i ) k } ) on C × S ˜ M . Let Y be the maximal locally closed subscheme of ˜ M such that ( l ( i ) k ) Y / ( l ( i ) k +1 ) Y is a locally free O D ( i ) Y -moduleof rank one for i = 1 , . . . , n and (cid:18) ˜ ∇| D ( i ) Y − ν ( i ) ( µ ( i ) k )id d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:19)(cid:0) (˜ l ( i ) k ) Y (cid:1) ⊂ ( l ( i ) k +1 ) Y ⊗ Ω C Y /Y ( D Y ) for1 ≤ k ≤ r . We set P := n (cid:89) i =1 Spec S ∗ Y (cid:16) H om ( ˜ E | D ( i ) Y , ˜ E | D ( i ) Y ) ∨ (cid:17) and take universal families ˜ N ( i ) : ˜ E | D ( i ) P −→ ˜ E | D ( i ) P for i = 1 , . . . , n , where S ∗ Y (cid:16) H om ( ˜ E | D ( i ) Y , ˜ E | D ( i ) Y ) ∨ (cid:17) denotes the symmetric algebra of H om ( ˜ E | D ( i ) Y , ˜ E | D ( i ) Y ) ∨ over Y . Let Z be the maximal locally closedsubscheme of P satisfying ϕ ˜ µ ( ˜ N ( i ) ) Z = 0 ∈ End( ˜ E | D ( i ) Z ), ˜ ν ( i ) ( ˜ N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( i ) Z = ˜ ∇| D ( i ) Z and O D ( i ) p [ T ] / ( ϕ ( i )˜ µ ( T )) (cid:51) P ( T ) (cid:55)→ P (( ˜ N ( i ) ) p ) ∈ End( ˜ E | D ( i ) p )is injective for any C -valued point p of Z . By construction, we can easily see that Z descends to a quasi-projective scheme M α C,D (˜ ν , ˜ µ ) over M , which is the desired moduli space.The proof of Theorem 2.11 will be completed at the end of subsection 2.7.2.3. Factorized ( ν , µ ) -connection. For the rest of the proof of Theorem 2.11, we need to describe thetangent space of the moduli space. We will describe the tangent space and give a symplectic structure viathe idea in section 1. So we introduce the notion of factorized ( ν , µ )-connection which comes from the ideaof factorization of a linear map in subsection 1.1.Let C, D, D ( i ) , D ( i ) j , µ , ϕ ( i ) µ , ν , ¯ z ( i ) and ¯ z ( i ) j be as in Definition 2.3. The following notion of factorizedconnection is useful for describing the deformation theory of ( ν , µ )-connections and the relative symplecticform on the moduli space. Definition 2.13. We say that a tuple ( E, ∇ , { θ ( i ) , κ ( i ) } ) is a factorized ( ν , µ )-connection if(1) E is an algebraic vector bundle on C of rank r and degree a ,(2) ∇ : E −→ E ⊗ Ω C ( D ) is an algebraic connection admitting poles along D ,(3) θ ( i ) : E ∨ | D ( i ) ∼ −→ E | D ( i ) is an O D ( i ) -isomorphism satisfying t θ ( i ) = θ ( i ) ,(4) κ ( i ) : E | D ( i ) −→ E ∨ | D ( i ) is an O D ( i ) -homomorphism satisfying t κ ( i ) = κ ( i ) ,(5) the composition N ( i ) := θ ( i ) ◦ κ ( i ) : E | D ( i ) −→ E | D ( i ) satisfies ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) , ϕ ( i ) µ ( N ( i ) ) = 0 and the injectivity of the ring homomorphism O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) ∈ End O D ( i ) ( E | D ( i ) ) . Two factorized ( ν , µ )-connections ( E, ∇ , { θ ( i ) , κ ( i ) } ) and ( E (cid:48) , ∇ (cid:48) , { θ (cid:48) ( i ) , κ (cid:48) ( i ) } ) are isomorphic if there is anisomorphism σ : E ∼ −→ E (cid:48) of algebraic vector bundles such that ( σ ⊗ ◦ ∇ = ∇ (cid:48) ◦ σ , and the diagrams E | D ( i ) κ ( i ) −−−−→ E ∨ | D ( i ) σ | D ( i ) (cid:121) ∼ = ∼ = (cid:121) t P ( i ) ( N ( i ) ) ◦ ( t σ | D ( i ) ) − E (cid:48) | D ( i ) κ (cid:48) ( i ) −−−−→ E (cid:48)∨ | D ( i ) E ∨ | D ( i ) θ ( i ) −−−−→ E | D ( i ) t P ( i ) ( N ( i ) ) ◦ ( t σ | D ( i ) ) − (cid:121) ∼ = σ | D ( i ) (cid:121) ∼ = E (cid:48)∨ | D ( i ) θ (cid:48) ( i ) −−−−→ E (cid:48) | D ( i ) are commutative for some P ( i ) ( T ) ∈ (cid:16) O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) (cid:17) × . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 17 Proposition 2.14. The correspondence ( E, ∇ , { θ ( i ) , κ ( i ) } ) (cid:55)→ ( E, ∇ , { θ ( i ) ◦ κ ( i ) } ) gives a bijective corre-spondence between the isomorphism classes of factorized ( ν , µ ) -connections and the isomorphism classes of ( ν , µ ) -connections on ( C, D ) .Proof. We will give the inverse correspondence. Let ( E, ∇ , { N ( i ) } ) be a ( ν , µ )-connection on ( C, D ). Wecan define an O D ( i ) [ T ]-module structure on E | D ( i ) by O D ( i ) [ T ] × E | D ( i ) (cid:51) ( P ( T ) , v ) (cid:55)→ P ( N ( i ) ) v ∈ E | D ( i ) . We also define an O D ( i ) [ T ]-module structure on E ∨ | D ( i ) by O D ( i ) [ T ] × E ∨ | D ( i ) (cid:51) ( P ( T ) , v ) (cid:55)→ P ( t N ( i ) ) v ∈ E ∨ | D ( i ) . For any point x ∈ D ( i ) , the homomorphism C [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) | x ) ∈ End C ( E | x ) is injectiveby Remark 2.4. So the minimal polynomial of the endomorphism N ( i ) | x on the vector space E | x is ϕ ( i ) µ | x ( T )whose degree is r = dim C E | x . Thus an elementary theory of linear algebra implies that there is an element v x ∈ E | x such that the homomorphism C [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) v x ∈ E | x is an isomorphism of C [ T ]-modules. If we take an element v ∈ E | D ( i ) such that v | x = v x for any x ∈ D ( i ) , then the homomorphism O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) v ∈ E | D ( i ) is an isomorphism of O D ( i ) [ T ]-modules. Similarly E ∨ | D ( i ) is isomorphic to O D ( i ) [ T ] / ( ϕ ( i ) µ ( T )) as an O D ( i ) [ T ]-module. So we can take an O D ( i ) [ T ]-isomorphism θ ( i ) : E ∨ | D ( i ) ∼ −→ E | D ( i ) , which makes the diagram E ∨ | D ( i ) θ ( i ) −−−−→ ∼ E | D ( i ) t N ( i ) (cid:121) N ( i ) (cid:121) E ∨ | D ( i ) θ ( i ) −−−−→ ∼ E | D ( i ) commutative. If we define κ ( i ) := ( θ ( i ) ) − ◦ N ( i ) : E | D ( i ) −→ E ∨ | D ( i ) , then κ ( i ) also becomes a homomorphism of O D ( i ) [ T ]-modules. By definition, we have θ ( i ) ◦ κ ( i ) = N ( i ) andwe can verify the equalities t θ ( i ) = θ ( i ) and t κ ( i ) = κ ( i ) in the same way as Proposition 1.1. We can see bythe same argument as Proposition 1.2 that the ambiguity of the choice of θ ( i ) is just a composition with theautomorphism of E | ∨D ( i ) s of the form P ( t N ( i ) ) for some P ( T ) ∈ C [ T ]. Thus we can define a correspondence( E, ∇ , { N ( i ) } ) (cid:55)→ ( E, ∇ , { θ ( i ) , κ ( i ) } ) which is the desired inverse correspondence by its construction. (cid:3) We extend the above proposition to a relative setting over a noetherian local scheme, that is, a schemeisomorphic to Spec A for some noetherian local ring A . Let C , D , D ( i ) , D ( i ) j , ˜ ν , ˜ µ , ϕ ( i )˜ µ , ¯ z ( i ) and ¯ z ( i ) j be as insubsection 2.2. Assume that S (cid:48) := Spec A (cid:48) is an noetherian local scheme with a morphism S (cid:48) −→ S . Wesay that ( E, ∇ , { N ( i ) } ) is a flat family of (˜ ν S (cid:48) , ˜ µ S (cid:48) )-connections on ( C S (cid:48) , D S (cid:48) ) over S (cid:48) if E is a vector bundleon C S (cid:48) of rank r , ∇ : E −→ E ⊗ Ω C S (cid:48) /S (cid:48) ( D S (cid:48) ) is an S (cid:48) -relative connection and N ( i ) : E | D ( i ) S (cid:48) −→ E | D ( i ) S (cid:48) is an O D ( i ) S (cid:48) -homomorphism such that ϕ ( i )˜ µ ( N ( i ) ) = 0, ˜ ν ( i ) ( N ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) S (cid:48) and the homomorphism O D ( i ) S (cid:48) [ T ] / ( ϕ ( i )˜ µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( N ( i ) ) ∈ End( E | D ( i ) S (cid:48) )is an injection whose cokernel is flat over S (cid:48) . Similarly we say that ( E, ∇ , { θ ( i ) , κ ( i ) } ) is a flat fam-ily of factorized (˜ ν S (cid:48) , ˜ µ S (cid:48) )-connections on ( C S (cid:48) , D S (cid:48) ) over S (cid:48) if E is a vector bundle on C S (cid:48) of rank r , ∇ : E −→ E ⊗ Ω C S (cid:48) /S (cid:48) ( D S (cid:48) ) is an S (cid:48) -relative connection, θ ( i ) : E ∨ | D ( i ) S (cid:48) −→ E | D ( i ) S (cid:48) is an isomorphism, κ ( i ) : E | D ( i ) S (cid:48) −→ E ∨ | D ( i ) S (cid:48) is a homomorphism such that t θ ( i ) = θ ( i ) , t κ ( i ) = κ ( i ) , ϕ ( i ) ( θ ( i ) ◦ κ ( i ) ) = 0, ν ( i ) ( θ ( i ) ◦ κ ( i ) ) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇| D ( i ) S (cid:48) and the homomorphism O D ( i ) S (cid:48) [ T ] / ( ϕ ( i )˜ µ ( T )) (cid:51) P ( T ) (cid:55)→ P ( θ ( i ) ◦ κ ( i ) ) ∈ End( E | D ( i ) S (cid:48) )is an injection whose cokernel is flat over S (cid:48) . Proposition 2.15. Let C , D , D ( i ) , D ( i ) j , ˜ ν , ˜ µ , ϕ ( i )˜ µ , ¯ z ( i ) and ¯ z ( i ) j be as in subsection 2.2 and let S (cid:48) be a noe-therian local scheme with a morphism S (cid:48) −→ S . Then the correspondence ( E, ∇ , { θ ( i ) , κ ( i ) } ) (cid:55)→ ( E, ∇ , { θ ( i ) ◦ κ ( i ) } ) gives a bijective correspondence between the flat families of factorized (˜ ν S (cid:48) , ˜ µ S (cid:48) ) -connections on ( C S (cid:48) , D S (cid:48) ) over S (cid:48) and the flat families of (˜ ν S (cid:48) , ˜ µ S (cid:48) ) -connections on ( C S (cid:48) , D S (cid:48) ) over S (cid:48) .Proof. The proof is exactly the same as that of Proposition 2.14. (cid:3) Tangent space of the moduli space of (˜ ν , ˜ µ ) -connections. We use the same notations as in sub-section 2.2. We take a C -valued point x of M α C , D (˜ ν , ˜ µ ) over a C -valued point s of S . Let (cid:0) E, ∇ , { N ( i ) } (cid:1) bethe ( ν , µ )-connection on the fiber ( C s , D s ) corresponding to x , where we put ( ν , µ ) := (˜ ν s , ˜ µ s ). By Propo-sition 2.14, we can take a factorized ( ν , µ )-connection (cid:0) E, ∇ , { θ ( i ) , κ ( i ) } (cid:1) corresponding to (cid:0) E, ∇ , { N ( i ) } (cid:1) .We will consider the deformation theory of (cid:0) E, ∇ , { N ( i ) } (cid:1) .Recall that ˜ ν ( i ) ( T ) is given by˜ ν ( i ) ( T ) = r − (cid:88) j =0 c ( i ) j T j ∈ H ( D ( i ) , O D ( i ) )[ T ] . We define homomorphisms σ ( i ) − θ ( i ) : End( E | D ( i ) s ) ⊕ O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) −→ Hom( E | ∨D ( i ) s , E | D ( i ) s ) σ ( i )+ κ ( i ) : End( E | D ( i ) s ) ⊕ O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) −→ Hom( E | D ( i ) s , E | ∨D ( i ) s ) δ ( i ) ν ,N ( i ) : End( E | D ( i ) s ) −→ End( E | D ( i ) s ) ⊗ Ω C s ( D s )by setting σ ( i ) − θ ( i ) (cid:0) u, P ( T ) (cid:1) = − u ◦ θ ( i ) − θ ( i ) ◦ t u + θ ( i ) ◦ P ( t N ( i ) )(15) σ ( i )+ κ ( i ) (cid:0) u, P ( T ) (cid:1) = κ ( i ) ◦ u + t u ◦ κ ( i ) − P ( t N ( i ) ) ◦ κ ( i ) (16) δ ( i ) ν ,N ( i ) ( u ) = r − (cid:88) j =1 j (cid:88) l =1 c ( i ) j ( N ( i ) ) j − l ◦ u ◦ ( N ( i ) ) l − d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (17)for u ∈ End( E | D ( i ) s ) and P ( T ) ∈ O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) . For each fixed u ∈ End( E | D ( i ) s ), we define a homo-morphism Θ ( i ) u : O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) −→ Ω C s ( D s ) (cid:12)(cid:12) D ( i ) s by setting(18) Θ ( i ) u (cid:0) P ( T ) (cid:1) = Tr (cid:0) P ( N ( i ) ) ◦ u (cid:1) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i for P ( T ) ∈ O D ( i ) s [ T ] / ( ϕ ( i ) µ ( T )). We put G := E nd ( E ) , G := E nd ( E ) ⊗ Ω C s ( D s ) , G := n (cid:77) i =1 Hom (cid:0) E | D ( i ) s , E | D ( i ) s ⊗ Ω C s ( D ( i ) s ) (cid:1) . Furthermore we put S ( E | ∨D s , E | D s ) = (cid:110) ( τ ( i ) ) ∈ n (cid:77) i =1 Hom (cid:0) E | ∨D ( i ) s , E | D ( i ) s (cid:1)(cid:12)(cid:12)(cid:12) t τ ( i ) = τ ( i ) for any i (cid:111) S ( E | D s , E | ∨D s ) = (cid:110) ( ξ ( i ) ) ∈ n (cid:77) i =1 Hom (cid:0) E | D ( i ) s , E | ∨D ( i ) s (cid:1)(cid:12)(cid:12)(cid:12) t ξ ( i ) = ξ ( i ) for any i (cid:111) and Z := n (cid:77) i =1 O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) , Z := n (cid:77) i =1 Hom O D ( i ) s (cid:16) O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) , Ω C s ( D ( i ) s ) | D ( i ) s (cid:17) . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 19 We define sheaves F , F , F on C s by F := G ⊕ Z , F := G ⊕ S ( E | ∨D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ) , F := G ⊕ Z and define homomorphisms d : F −→ F , d : F −→ F by d (cid:0) u, ( P ( i ) ( T )) (cid:1) = (cid:16) ∇ ◦ u − u ◦ ∇ , (cid:16) σ ( i ) − θ ( i ) (cid:0) u | D ( i ) s , P ( i ) ( T ) (cid:1)(cid:17) , (cid:16) σ ( i )+ κ ( i ) (cid:0) u | D ( i ) s , P ( i ) ( T ) (cid:1)(cid:17)(cid:17) d (cid:0) v, ( τ ( i ) ) , ( ξ ( i ) ) (cid:1) = (cid:16)(cid:16) v | D ( i ) s − δ ( i ) ν ,N ( i ) (cid:0) τ ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) (cid:1)(cid:17) , (cid:16) Θ ( i )( τ ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) ) (cid:17)(cid:17) . Lemma 2.16. Under the above notation, d ◦ d = 0 .Proof. Take ( u, ( P ( i ) ( T ))) ∈ F = G ⊕ Z . Note that σ ( i ) − θ ( i ) (cid:0) u | D ( i ) s , ( P ( i ) ( T )) (cid:1) ◦ κ ( i ) + θ ( i ) ◦ σ ( i )+ κ ( i ) (cid:0) u | D ( i ) s , ( P ( i ) ( T )) (cid:1) = (cid:16) − u | D ( i ) s ◦ θ ( i ) − θ ( i ) ◦ t u | D ( i ) s + θ ( i ) ◦ P ( i ) ( t N ( i ) ) (cid:17) ◦ κ ( i ) + θ ( i ) ◦ (cid:16) κ ( i ) ◦ u | D ( i ) s + t u | D ( i ) s ◦ κ ( i ) − P ( i ) ( t N ( i ) ) ◦ κ ( i ) (cid:17) = θ ( i ) ◦ κ ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ θ ( i ) ◦ κ ( i ) = N ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ N ( i ) . So the first component of d (cid:0) d (cid:0) u, ( P ( i ) ( T )) (cid:1)(cid:1) is (cid:16) ( ∇ ◦ u − u ◦ ∇ ) | D ( i ) s − δ ( i ) ν ,N ( i ) (cid:16) σ ( i ) − θ ( i ) (cid:0) u | D ( i ) s , ( P ( i ) ( T )) (cid:1) ◦ κ ( i ) + θ ( i ) ◦ σ ( i )+ κ ( i ) (cid:0) u | D ( i ) s , ( P ( i ) ( T )) (cid:1)(cid:17)(cid:17) = (cid:16) ( ∇ ◦ u − u ◦ ∇ ) | D ( i ) s − δ ( i ) ν ,N ( i ) ( N ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ N ( i ) ) (cid:17) = ( ∇ ◦ u − u ◦ ∇ ) | D ( i ) s − r − (cid:88) j =1 j (cid:88) l =1 c ( i ) j ( N ( i ) ) j − l ◦ ( N ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ N ( i ) ) ◦ ( N ( i ) ) l − d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = ( ∇ ◦ u − u ◦ ∇ ) | D ( i ) s − (cid:16) r − (cid:88) j =0 c ( i ) j ( N ( i ) ) j ◦ u | D ( i ) s − r − (cid:88) j =0 c ( i ) j u | D ( i ) s ◦ ( N ( i ) ) j (cid:17) d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = (cid:16) ( ∇ ◦ u − u ◦ ∇ ) | D ( i ) s − (cid:0) ∇| D ( i ) s ◦ u | D ( i ) s − u | D ( i ) s ◦ ∇| D ( i ) s (cid:1)(cid:17) = 0 . The second component of d (cid:0) d (cid:0) u, ( P ( i ) ( T )) (cid:1)(cid:1) is (cid:32) Θ ( i ) σ ( i ) − θ ( i ) ( u | D ( i ) s , ( P ( i ) ( T ))) ◦ κ ( i ) + θ ( i ) ◦ σ ( i )+ κ ( i ) ( u | D ( i ) s , ( P ( i ) ( T ))) (cid:33) = (cid:18) Θ ( i ) N ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ N ( i ) (cid:19) , which is zero becauseΘ ( i ) N ( i ) ◦ u | D ( i ) s − u | D ( i ) s ◦ N ( i ) ( Q ( T ))= Tr (cid:16) Q ( N ( i ) ) ◦ N ( i ) ◦ u | D ( i ) s − Q ( N ( i ) ) ◦ u | D ( i ) s ◦ N ( i ) (cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = (cid:16) Tr (cid:16) Q ( N ( i ) ) ◦ N ( i ) ◦ u | D ( i ) s (cid:17) − Tr (cid:16) N ( i ) ◦ Q ( N ( i ) ) ◦ u | D ( i ) s (cid:17)(cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = 0for any Q ( T ) ∈ O D ( i ) s [ T ] (cid:14)(cid:0) ϕ ( i ) µ ( T ) (cid:1) . Thus we have proved d (cid:0) d (cid:0) u, ( P ( i ) ( T )) (cid:1)(cid:1) = 0. (cid:3) By Lemma 2.16, F • = [ F d −→ F d −→ F ] becomes a complex. Note that there is an exact commutativediagram0 −→ −−−−→ G ⊕ Z −−−−→ G ⊕ Z −→ (cid:121) d (cid:121) (cid:121) −→G ⊕ S ( E | D s , E | ∨D s ) −−−−→ G ⊕ S ( E | ∨D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ) −−−−→ S ( E | ∨D s , E | D s ) −→ (cid:121) d (cid:121) (cid:121) −→ G ⊕ Z −−−−→ G ⊕ Z −−−−→ −→ . If we denote by F • the complex G ⊕ Z −→ S ( E | ∨D s , E | D s ) concentrated in degree 0 and 1 and if wedenote by F • the complex G ⊕ S ( E | D s , E | ∨D s ) −→ G ⊕ Z concentrated in degree 0 and 1, then the abovecommutative diagram is a short exact sequence of complexes(19) 0 −→ F • [ − −→ F • −→ F • −→ −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ . Proposition 2.17. Let A be an artinian local ring over S with the maximal ideal m satisfying A/ m = C and let I be an ideal of A satisfying m I = 0 . Assume that there exists a flat family ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ∈M α C,D (˜ ν , ˜ µ )( A ) of (˜ ν , ˜ µ ) -connections over A such that ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ⊗ A/ m ∼ = ( E, ∇ , { N ( i ) } ) . Considerthe restriction map ρ A/I : M α C,D (˜ ν , ˜ µ )( A ) (cid:51) ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) (cid:55)→ ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) ⊗ A/I ∈ M α C,D (˜ ν , ˜ µ )( A/I ) . Then there exists a bijective correspondence ρ − A/I (( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ⊗ A/I ) ∼ = H ( F • ) ⊗ C I .Proof. We can take an affine open covering C A = (cid:83) α U α such that (cid:93) { i | D ( i ) A ∩ U α (cid:54) = ∅} ≤ α and (cid:93) { α | D ( i ) A ⊂ U α } = 1 for any i . We may assume that E (cid:48) | U α ∼ = O ⊕ rU α for any α . Take any member( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) ∈ ρ − A/I (( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ⊗ A/I ). Let ( E (cid:48) , ∇ (cid:48) , { θ (cid:48) ( i ) , κ (cid:48) ( i ) } ) and ( ˜ E, ˜ ∇ , { ˜ θ ( i ) , ˜ κ ( i ) } ) be the flatfamilies of factorized (˜ ν , ˜ µ ) ⊗ A -connections on ( C A , D A ) over A corresponding to ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) and( ˜ E, ˜ ∇ , { ˜ N ( i ) } ), respectively. We can take an isomorphism σ α : ˜ E | U α ∼ −→ E (cid:48) | U α which is a lift of the givenisomorphism ˜ E ⊗ A/I | U α ⊗ A/I ∼ −→ E (cid:48) ⊗ A/I | U α ⊗ A/I . Then we put u αβ := σ α ◦ σ − β − id E (cid:48) | Uαβ ∈ G ( U αβ ) ⊗ I, v α := σ α ◦ ˜ ∇ ◦ σ − α − ∇ (cid:48) ∈ G ( U α ) ⊗ I and τ ( i ) α := σ α | D ( i ) A ◦ ˜ θ ( i ) ◦ t σ α | D ( i ) A − θ (cid:48) ( i ) , ξ ( i ) α := t σ α | − D ( i ) A ◦ ˜ κ ( i ) ◦ σ α | − D ( i ) A − κ (cid:48) ( i ) if D ( i ) A ⊂ U α . Note that we have (( τ ( i ) α ) , ( ξ ( i ) α )) ∈ ( S ( E | ∨D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ))( U α ) ⊗ C I . We can easilycheck the equalities u βγ − u αγ + u αβ = 0 , ∇ ◦ u αβ − u αβ ◦ ∇ = v β − v α . Since τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α = (cid:0) σ α | D ( i ) A ◦ ˜ θ ( i ) ◦ t σ α | D ( i ) A − θ (cid:48) ( i ) (cid:1) ◦ t σ α | − D ( i ) A ◦ ˜ κ ( i ) ◦ σ α | − D ( i ) A + θ (cid:48) ( i ) ◦ (cid:0) t σ α | − D ( i ) A ◦ ˜ κ ( i ) ◦ σ α | − D ( i ) A − κ (cid:48) ( i ) (cid:1) = σ α | D ( i ) A ◦ ˜ θ ( i ) ◦ ˜ κ ( i ) ◦ σ α | − D ( i ) A − θ (cid:48) ( i ) ◦ κ (cid:48) ( i ) = σ α | D ( i ) A ◦ ˜ N ( i ) ◦ σ α | − D ( i ) A − N (cid:48) ( i ) , NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 21 we have δ ( i ) ν ,N ( i ) ( τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α )= r − (cid:88) j =1 j (cid:88) l =1 c ( i ) j (cid:0) σ α | D ( i ) A ◦ ˜ N ( i ) ◦ σ α | − D ( i ) A (cid:1) j − l ◦ (cid:0) σ α | D ( i ) A ◦ ˜ N ( i ) ◦ σ α | − D ( i ) A − N (cid:48) ( i ) (cid:1) ◦ ( N (cid:48) ( i ) ) l − d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = r − (cid:88) j =1 c ( i ) j (cid:16)(cid:0) σ α | D ( i ) A ◦ ˜ N ( i ) ◦ σ α | − D ( i ) A (cid:1) j − (cid:0) N (cid:48) ( i ) (cid:1) j (cid:17) d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = σ α | D ( i ) A ◦ ˜ ∇| D ( i ) A ◦ σ α | − D ( i ) A − ∇ (cid:48) | D ( i ) A . So the first component of d ( v α , ( τ ( i ) α ) , ( ξ ( i ) α )) becomes v α | D ( i ) A − δ ( i ) ν ,N ( i ) ( τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α ) = ( σ α ◦ ˜ ∇ ◦ σ − α − ∇ (cid:48) ) | D ( i ) A − ( σ α | D ( i ) A ◦ ˜ ∇| D ( i ) A ◦ σ α | − D ( i ) A − ∇ (cid:48) | D ( i ) A )= 0 . On the other hand, N (cid:48) ( i ) has a representation matrix µ ( i )1 · · · ... . . . ... · · · µ ( i ) r with respect to a basis e (cid:48) , . . . , e (cid:48) r of E (cid:48) | D ( i ) A and ˜ N ( i ) has the same representation matrix with respect toa basis ˜ e , . . . , ˜ e r of ˜ E | D ( i ) A from Definition 2.10, (c). Moreover, we may assume that ( e (cid:48) , . . . , e (cid:48) r ) ⊗ A/I =(˜ e , . . . , ˜ e r ) ⊗ A/I , because ˜ N ( i ) ⊗ A/I = N (cid:48) ( i ) ⊗ A/I . So there exists g ∈ I End( E (cid:48) | D ( i ) A ) satisfying(id − g ) ◦ N (cid:48) ( i ) ◦ (id + g ) = σ α | D ( i ) A ◦ ˜ N ( i ) ◦ t σ α | D ( i ) A . In other words, σ α | D ( i ) A ◦ ˜ N ( i ) ◦ t σ α | D ( i ) A − N (cid:48) ( i ) = N (cid:48) ( i ) ◦ g − g ◦ N (cid:48) ( i ) = N ( i ) ◦ g − g ◦ N ( i ) . So the second component of d ( v α , ( τ ( i ) α ) , ( ξ ( i ) α )) becomesΘ ( i )( τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α ) = Θ ( i )( σ α | D ( i ) A ◦ ˜ N ( i ) ◦ σ α | − D ( i ) A − N (cid:48) ( i ) ) = Θ ( i )( N ( i ) ◦ g − g ◦ N ( i ) ) = 0 . Thus the element Φ( v ) := (cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , ( τ ( i ) α ) , ( ξ ( i ) α )) (cid:9)(cid:3) ∈ H ( F • ) ⊗ I can be defined.Conversely assume that w = (cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , ( τ ( i ) α ) , ( ξ ( i ) α )) (cid:9)(cid:3) ∈ H ( F • ) ⊗ I is given. We put E α := E (cid:48) | U α and define a connection ∇ α : E α −→ E α ⊗ Ω C A /A ( D A ) by ∇ α = ∇ (cid:48) + v α . Furthermore, we put θ ( i ) α := θ (cid:48) ( i ) + τ ( i ) α , κ ( i ) α := κ (cid:48) ( i ) + ξ ( i ) α if D ( i ) A ⊂ U α . We define the isomorphism ϕ βα = id + u βα : E α | U αβ ∼ −→ E β | U αβ . Since ( { ( u αβ , } , { ( v α , ( τ ( i ) α ) , ( ξ ( i ) α )) } ) satisfies the cocycle conditions ∇ ◦ u αβ − u αβ ◦ ∇ = v β − v α and u βα − u γα + u γβ = 0, we have the gluing condition ϕ γα = ϕ γβ ◦ ϕ βα , ( ϕ βα ⊗ ◦ ∇ α = ∇ β ◦ ϕ βα . So we can patch the local connections { ( E α , ∇ α , { θ ( i ) α , κ ( i ) α } ) } together via { ϕ βα } and obtain a flat family( ˜ E, ˜ ∇ , { ˜ θ ( i ) , ˜ κ ( i ) } ) of factorized (˜ ν , ˜ µ ) ⊗ A -connections over A , which we denote by Ψ( w ). By constructionthe correspondence H ( F • ) ⊗ I (cid:51) w (cid:55)→ Ψ( w ) ∈ ρ − A/I (( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ⊗ A/I ) gives the inverse of Φ. (cid:3) As a corollary of Proposition 2.17, we get the following. Corollary 2.18. The relative tangent space of the moduli space M α C , D (˜ ν , ˜ µ ) over S at ( E, ∇ , { N ( i ) } ) ∈ M α C , D (˜ ν , ˜ µ ) is isomorphic to H ( F • ) . Nondegenerate pairing on the cohomologies. We use the same notations as in subsection 2.4.If we denote the complex O C d −→ Ω C /S ( D ) −→ Ω C /S ( D ) | D . by L • , then there is a canonical quasi-isomorphism Ω •C /S −→ L • and there is an isomorphism H ( L • s ) ∼ = H (Ω •C s ) ∼ = C , where L • s := L • | C s is the restriction of the complex L • to the fiber C s . We consider the modified complex˜ L • s : L s ˜ d −→ L s ⊕ Z d −→ L s ⊕ Z , defined by ˜ d ( u ) = ( du, , ˜ d ( v, ( Q i )) = (cid:16)(cid:0) v | D ( i ) s − Q i (cid:0) ( ν ( i ) ) (cid:48) ( T ) (cid:1)(cid:1) , ( Q i ) (cid:17) , where ( ν ( i ) ) (cid:48) ( T ) is the derivative of the polynomial ν ( i ) ( T ) in T . Then there is a canonical quasi-isomorphism L • s −→ ˜ L • s .We define a morphism of complexes Tr : F • −→ ˜ L • s byTr (cid:0) u, ( P ( i ) ( T )) (cid:1) = Tr( u ) , Tr (cid:0) v, ( τ ( i ) ) , ( ξ ( i ) ) (cid:1) = (cid:0) Tr( v ) , (Θ τ ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) ) (cid:1) , Tr (cid:0) ( g ( i ) ) , ( Q ( i ) ) (cid:1) = (cid:0) (Tr( g ( i ) )) , ( Q ( i ) ) (cid:1) . Indeed we can check the following commutative diagram: G ⊕ Z −−−−→ G ⊕ S ( E | ∨D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ) −−−−→ G ⊕ Z (cid:121) Tr (cid:121) Tr (cid:121) O C s d −−−−→ Ω C s ( D s ) ⊕ Z −−−−→ Ω C s ( D s ) | D s ⊕ Z . For (( τ ( i ) ) , ( ξ ( i ) )) , (( τ (cid:48) ( i ) ) , ( ξ (cid:48) ( i ) )) ∈ S ( E | ∨D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ), we define Ξ ( τ ( i ) ,ξ ( i ) )( τ (cid:48) ( i ) ,ξ (cid:48) ( i ) ) ∈ Ω C s ( D ( i ) s ) | D ( i ) s by settingΞ ( τ ( i ) ,ξ ( i ) )( τ (cid:48) ( i ) ,ξ (cid:48) ( i ) ) = 12 r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:16) τ (cid:48) ( i ) ◦ ( t N ( i ) ) l ◦ ξ ( i ) ◦ ( N ( i ) ) j − l − (cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (21) − r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:16) τ ( i ) ◦ ( t N ( i ) ) l ◦ ξ (cid:48) ( i ) ◦ ( N ( i ) ) j − − l (cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i . Remark 2.19. In the extreme case when µ ( i ) k = ν ( i ) ( µ ( i ) k ) for any k , we have c ( i )1 = 1 and c ( i ) j = 0 for j (cid:54) = 1.So we have Ξ ( τ ( i ) ,ξ ( i ) )( τ (cid:48) ( i ) ,ξ (cid:48) ( i ) ) = 12 Tr (cid:16) τ (cid:48) ( i ) ◦ ξ ( i ) − τ ( i ) ◦ ξ (cid:48) ( i ) (cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i which is almost the same form as the expression in subsection 1.2, (3) of the Kirillov-Kostant form inProposition 1.5.We define a bilinear pairing ω ( E, ∇ , { N ( i ) } ) : H ( F • ) × H ( F • ) −→ H ( L • s ) ∼ = C on H ( F • ) by setting ω ( E, ∇ , { N ( i ) } ) (cid:16)(cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , (( τ ( i ) α ) , ( ξ ( i ) α ))) (cid:9)(cid:3) , (cid:2)(cid:8) ( u (cid:48) αβ , (cid:9) , (cid:8) ( v (cid:48) α , (( τ (cid:48) ( i ) α ) , ( ξ (cid:48) ( i ) α ))) (cid:9)(cid:3)(cid:17) (22) = (cid:104)(cid:8) Tr( u αβ ◦ u (cid:48) βγ ) (cid:9) , − (cid:8) Tr( u αβ ◦ v (cid:48) β − v α ◦ u (cid:48) αβ ) (cid:9) , (cid:110)(cid:16) Ξ ( τ ( i ) α ,ξ ( i ) α )( τ (cid:48) ( i ) α ,ξ (cid:48) ( i ) α ) (cid:17)(cid:111)(cid:105) ∈ H ( L • s ) . We will check that the cohomology class (22) in H ( L • s ) is independent of the choice of the representatives (cid:0)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , (( τ ( i ) α ) , ( ξ ( i ) α ))) (cid:9)(cid:1) and (cid:0)(cid:8) ( u (cid:48) αβ , (cid:9) , (cid:8) ( v (cid:48) α , (( τ (cid:48) ( i ) α ) , ( ξ (cid:48) ( i ) α ))) (cid:9)(cid:1) , respectively. Indeed assume NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 23 that (cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , (( τ ( i ) α ) , ( ξ ( i ) α ))) (cid:9)(cid:3) = 0 in H ( F • ). Then there is (cid:8) u α , (cid:0) P ( i ) α ( T ) (cid:1)(cid:9) ∈ C ( { U α } , G ⊕ Z )satisfying u αβ = u β − u α , v α = ∇ ◦ u α − u α ◦ ∇ ,τ ( i ) α = σ ( i ) − θ ( i ) ( u α | D ( i ) s , P ( i ) ( T )) = − ( u α | D ( i ) s ◦ θ ( i ) + θ ( i ) ◦ t u α | D ( i ) s ) + θ ( i ) ◦ P ( t N ( i ) ) ξ ( i ) α = σ ( i ) + κ ( i ) ( u α | D ( i ) s , P ( i ) ( T )) = κ ( i ) ◦ u α | D ( i ) s + t u α | D ( i ) s ◦ κ ( i ) − P ( t N ( i ) ) ◦ κ ( i ) . So we can write ω ( E, ∇ , { N ( i ) } ) (cid:16)(cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , (( τ ( i ) α ) , ( ξ ( i ) α ))) (cid:9)(cid:3) , (cid:2)(cid:8) ( u (cid:48) αβ , (cid:9) , (cid:8) ( v (cid:48) α , (( τ (cid:48) ( i ) α ) , ( ξ (cid:48) ( i ) α ))) (cid:9)(cid:3)(cid:17) (23) = (cid:20)(cid:8) Tr(( u β − u α ) ◦ u (cid:48) βγ ) (cid:9) , − (cid:8) Tr(( u β − u α ) ◦ v (cid:48) β − ( ∇ ◦ u α − u α ◦ ∇ ) ◦ u (cid:48) αβ ) (cid:9) , (cid:110)(cid:16) Ξ σ ( i ) − θ ( i ) ( u α | D ( i ) s ,P ( i ) ( T )) ,σ ( i )+ κ ( i ) ( u α | D ( i ) s ,P ( i ) ( T ))( τ (cid:48) ( i ) α ,ξ (cid:48) ( i ) α ) (cid:17)(cid:111)(cid:21) . If we put c αβ := Tr( u α ◦ u (cid:48) αβ ), then { c αβ } ∈ C ( { U α } , L s ) and(24) (cid:8) Tr(( u β − u α ) ◦ u (cid:48) βγ ) (cid:9) = (cid:8) Tr( u β ◦ u (cid:48) βγ − u α ◦ ( u (cid:48) αγ − u (cid:48) αβ )) (cid:9) = (cid:8) c βγ − c αγ + c αβ (cid:9) . If we put b α := Tr( u α ◦ v (cid:48) α ), then { b α } ∈ C ( { U α } , L s ) and we have d L • s (cid:0) { c αβ } (cid:1) = (cid:8) d Tr( u α ◦ u (cid:48) αβ ) (cid:9) = (cid:8) Tr( ∇ ◦ u α ◦ u (cid:48) αβ − u α ◦ u (cid:48) αβ ◦ ∇ ) (cid:9) (25) = (cid:8) Tr(( ∇ ◦ u α − u α ◦ ∇ ) ◦ u (cid:48) αβ + u α ◦ ( ∇ ◦ u (cid:48) αβ − u (cid:48) αβ ◦ ∇ )) (cid:9) = (cid:8) Tr(( ∇ ◦ u α − u α ◦ ∇ ) ◦ u (cid:48) αβ + u α ◦ ( v (cid:48) β − v (cid:48) α )) (cid:9) = (cid:8) Tr(( ∇ ◦ u α − u α ◦ ∇ ) ◦ u (cid:48) αβ + ( u α − u β ) ◦ v (cid:48) β + ( u β ◦ v (cid:48) β − u α ◦ v (cid:48) α )) (cid:9) = − (cid:8) Tr(( u β − u α ) ◦ v (cid:48) β − ( ∇ ◦ u α − u α ◦ ∇ ) ◦ u (cid:48) αβ ) (cid:9) + (cid:8) b β − b α (cid:9) . Since Tr (cid:0) ( τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α ) ◦ ( N ( i ) ) l ◦ P ( i ) ( N ( i ) ) ◦ ( N ( i ) ) j − − l (cid:1) = 0 follows from Θ ( i ) τ (cid:48) ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) = 0,Tr (cid:0) τ (cid:48) ( i ) α ◦ ( t N ( i ) ) l ◦ σ ( i ) + κ ( i ) ( u α | D ( i ) s , P ( i ) ( T )) ◦ ( N ( i ) ) j − − l (cid:1) − Tr (cid:0) σ ( i ) − θ ( i ) ( u α | D ( i ) s , P ( i ) ( T )) ◦ ( t N ( i ) ) l ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) j − − l (cid:1) = Tr (cid:16) τ (cid:48) ( i ) α ◦ ( t N ( i ) ) l ◦ (cid:0) κ ( i ) ◦ u α | D ( i ) s + t u α | D ( i ) s ◦ κ ( i ) − P ( i ) ( t N ( i ) ) ◦ κ ( i ) (cid:1) ◦ ( N ( i ) ) j − − l (cid:17) − Tr (cid:16)(cid:0) − u α | D ( i ) s ◦ θ ( i ) − θ ( i ) ◦ t u α | D ( i ) s + θ ( i ) ◦ P ( i ) ( t N ( i ) ) (cid:1) ◦ ( t N ( i ) ) l ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) j − − l (cid:17) = Tr (cid:16) τ (cid:48) ( i ) α ◦ ( t N ( i ) ) l ◦ κ ( i ) ◦ u α | D ( i ) s ◦ ( N ( i ) ) j − − l + ( t N ( i ) ) j − − l ◦ κ ( i ) ◦ u α | D ( i ) s ◦ ( N ( i ) ) l ◦ τ (cid:48) ( i ) α (cid:17) + Tr (cid:16) u α | D ( i ) s ◦ θ ( i ) ◦ ( t N ( i ) ) l ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) j − − l + ( t N ( i ) ) j − − l ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) l ◦ u α | D ( i ) s ◦ θ ( i ) (cid:17) − Tr (cid:16)(cid:0) τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α (cid:1) ◦ ( N ( i ) ) l ◦ P ( i ) ( N ( i ) ) ◦ ( N ( i ) ) j − − l (cid:17) = Tr (cid:16) u α | D ( i ) s ◦ ( N ( i ) ) j − − l ◦ τ (cid:48) ( i ) α ◦ κ ( i ) ◦ ( N ( i ) ) l + u α | D ( i ) s ◦ ( N ( i ) ) l ◦ τ (cid:48) ( i ) α ◦ κ ( i ) ◦ ( N ( i ) ) j − − l (cid:17) + Tr (cid:16) u α | D ( i ) s ◦ ( N ( i ) ) l ◦ θ ( i ) ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) j − − l + u α | D ( i ) s ◦ ( N ( i ) ) j − − l ◦ θ ( i ) ◦ ξ (cid:48) ( i ) α ◦ ( N ( i ) ) l (cid:17) . So we haveΞ σ ( i ) − θ ( i ) ( u α | D ( i ) s ,P ( i ) ( T )) ,σ ( i )+ κ ( i ) ( u α | D ( i ) s ,P ( i ) ( T ))( τ (cid:48) ( i ) α ,ξ (cid:48) ( i ) α ) = 12 r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:18) τ (cid:48) ( i ) α ◦ (cid:0) t N ( i ) (cid:1) l ◦ σ ( i ) + κ ( i ) (cid:0) u α | D ( i ) s , P ( i ) ( T ) (cid:1) ◦ (cid:0) N ( i ) (cid:1) j − − l − σ ( i ) − θ ( i ) (cid:0) u α | D ( i ) s , P ( i ) ( T ) (cid:1) ◦ ( t N ( i ) ) l ◦ ξ (cid:48) ( i ) α ◦ (cid:0) N ( i ) (cid:1) j − − l (cid:19) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:18) u α | D ( i ) s ◦ (cid:0) N ( i ) (cid:1) l ◦ (cid:0) τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α (cid:1) ◦ (cid:0) N ( i ) (cid:1) j − − l (cid:19) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = Tr (cid:16) u α | D ( i ) s ◦ δ ( i ) ν ,N ( i ) (cid:0) τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α (cid:1)(cid:17) . Since v (cid:48) α | D ( i ) s = δ ( i ) ν ,N ( i ) ( τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α ), we have d L • s { ( b α ) } = (cid:110)(cid:16) Tr( u α ◦ v (cid:48) α ) | D ( i ) s (cid:17)(cid:111) = (cid:110)(cid:16) Tr (cid:16) u α | D ( i ) s ◦ δ ( i ) ν ,N ( i ) ( τ (cid:48) ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ (cid:48) ( i ) α ) (cid:17)(cid:17)(cid:111) (26) = (cid:110)(cid:16) Ξ σ ( i ) − θ ( i ) ( u α | D ( i ) s ,P ( i ) ( T )) ,σ ( i )+ κ ( i ) ( u α | D ( i ) s ,P ( i ) ( T ))( τ (cid:48) ( i ) α ,ξ (cid:48) ( i ) α ) (cid:17)(cid:111) . The equalities (24), (25) and (26) mean that the cohomology class (23) is represented as the cobound-ary of (cid:0)(cid:8) c αβ (cid:9) , (cid:8) b α (cid:9)(cid:1) ∈ C ( { U α } , L • s ), which should be zero in H ( L • s ). Similarly (22) becomes zerowhen (cid:2)(cid:8) ( u (cid:48) αβ , (cid:9) , (cid:8) ( v (cid:48) α , ( τ (cid:48) ( i ) α ) , ( ξ (cid:48) ( i ) α )) (cid:9) ] = 0 in H ( F • ). Thus we have proved that the bilinear pairing ω ( E, ∇ , { N ( i ) } ) is well-defined. Lemma 2.20. The bilinear pairing ω ( E, ∇ , { N ( i ) } ) : H ( F • ) × H ( F • ) −→ H ( L • s ) ∼ = C defined in (22) is anon-degenerate pairing.Proof. Let σ : H ( F • ) −→ H ( F ) ∨ be the homomorphism determined by the pairing ω ( E, ∇ , { N ( i ) } ) . We haveto show that σ is an isomorphism. We can see that σ induces the following exact commutative diagram H ( F • ) −−−−→ H ( F • ) −−−−→ H ( F • ) −−−−→ H ( F • ) −−−−→ H ( F • ) σ (cid:121) σ (cid:121) σ (cid:121) σ (cid:121) σ (cid:121) H ( F • ) ∨ −−−−→ H ( F • ) ∨ −−−−→ H ( F • ) ∨ −−−−→ H ( F • ) ∨ −−−−→ H ( F • ) ∨ . Here σ : H ( F • ) −→ H ( F • ) ∨ and σ : H ( F • ) −→ H ( F • ) ∨ are given by the pairing H ( F • ) × H ( F • ) −→ H ( L • s ) ∼ = C (cid:16)(cid:104)(cid:8) ( v α , ( ξ ( i ) α )) (cid:9)(cid:105) , (cid:104)(cid:8) ( u (cid:48) αβ , ( τ (cid:48) ( i ) α )) (cid:9)(cid:105)(cid:17) (cid:55)→ (cid:104)(cid:8) Tr( v α ◦ u (cid:48) αβ ) (cid:9) , (cid:110)(cid:0) Ξ (0 ,ξ ( i ) α )( τ (cid:48) ( i ) α , (cid:1)(cid:111)(cid:105) and σ : H ( F • ) −→ H ( F • ) ∨ and σ : H ( F • ) −→ H ( F • ) ∨ are defined by the pairing H ( F • ) × H ( F • ) −→ H ( L • s ) ∼ = C (cid:16)(cid:2)(cid:8) ( u α , ( P ( i ) α )) (cid:9)(cid:3) , (cid:2)(cid:8) v (cid:48) αβ (cid:9) , (cid:8) ( g (cid:48) ( i ) α ) , ( Q (cid:48) ( i ) α ) (cid:9)(cid:3)(cid:17) (cid:55)→ (cid:34)(cid:8) − Tr( u α ◦ v (cid:48) αβ ) (cid:9) , − (cid:110)(cid:16) Tr (cid:0) u α | D ( i ) s ◦ g (cid:48) ( i ) α (cid:1) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i + 12 Q (cid:48) ( i ) α (cid:0) P ( i ) α ( T )( ν ( i ) ) (cid:48) ( T ) (cid:1)(cid:17)(cid:111)(cid:35) . We denote the short exact sequence of complexes0 −−−−→ G −−−−→ G ⊕ S ( E | D s , E | ∨D s ) −−−−→ S ( E | D s , E | ∨D s ) −−−−→ (cid:121) (cid:121) (cid:121) −−−−→ G −−−−→ G ⊕ Z −−−−→ Z −−−−→ NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 25 simply by 0 −→ [ G → G ] −→ F • −→ [ S ( E | D s , E | ∨D s ) → Z ] −→ −−−−→ Z −−−−→ G ⊕ Z −−−−→ G −−−−→ (cid:121) (cid:121) (cid:121) −−−−→ S ( E | ∨D s , E | D s ) −−−−→ S ( E | ∨D s , E | D s ) −−−−→ −−−−→ −→ [ Z → S ( E | ∨D s , E | D s )] −→ F • −→ G −→ 0. These short exact sequences of complexesinduce the exact commutative diagram0 −→ H (ker( G → G )) −→ H ( F • ) −→ ker( S ( E | D s , E | ∨D s ) → Z ) −→ H (ker( G → G )) η (cid:121) σ (cid:121) η (cid:121) η (cid:121) −→ H ( G ) ∨ −→ H ( F • ) ∨ −→ coker( Z → S ( E | ∨D s , E | D s )) ∨ −→ H ( G ) ∨ . Here η and η are induced by the trace pairing G ⊗ ker( G → G ) (cid:51) u ⊗ v (cid:55)→ Tr( u ⊗ v ) ∈ Ω C s and the isomorphism H (Ω C s ) ∼ −→ H ( ˜ L • s ) ∼ −→ C . Since the above trace pairing induces the isomorphismker( G → G ) ∼ −→ ( G ) ∨ ⊗ Ω C s , η , η are the isomorphisms induced by this isomorphism and the Serreduality. The homomorphism η is induced by the pairingker (cid:0) S ( E | D s , E | ∨D s ) → Z (cid:1) × coker (cid:0) Z → S ( E | ∨D s , E | D s ) (cid:1) −→ H ( L • s ) ∼ = C (27) (( ξ ( i ) ) , ( τ ( i ) )) (cid:55)→ (cid:104)(cid:110)(cid:0) Ξ (0 ,ξ ( i ) )( τ ( i ) , (cid:1)(cid:111)(cid:105) . Note that (cid:104)(cid:0) Ξ (0 ,ξ ( i ) )( τ ( i ) , (cid:1)(cid:105) ∈ H ( L • s ) corresponds to12 n (cid:88) i =1 res p ∈D ( i ) s (cid:18) r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:16) τ ( i ) ◦ ( t N ( i ) ) l ◦ ξ ( i ) ◦ ( N ( i ) ) j − l (cid:17) d ¯ z ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i (cid:19) via the isomorphism H ( L • s ) ∼ −→ C . Let us consider the restriction to each point p ∈ D s of the pairingker (cid:0) S ( E | D s , E | ∨D s ) → Z (cid:1) × coker (cid:0) Z → S ( E | ∨D s , E | D s ) (cid:1) −→ O D s (28) (( ξ ( i ) ) , ( τ ( i ) )) (cid:55)→ n (cid:88) i =1 r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:16) τ ( i ) ◦ ( t N ( i ) ) l ◦ ξ ( i ) ◦ ( N ( i ) ) j − − l (cid:17) . Assume that ( ξ ( i ) ) ∈ ker (cid:0) S ( E | D s , E | ∨D s ) → Z (cid:1) p satisfies n (cid:88) i =1 r − (cid:88) j =1 j − (cid:88) l =0 c ( i ) j Tr (cid:16) τ ( i ) ◦ ( t N ( i ) ) l ◦ ξ ( i ) ◦ ( N ( i ) ) j − − l (cid:17) = 0for any ( τ ( i ) ) ∈ coker (cid:0) Z → S ( E | ∨D s , E | D s ) (cid:1) p . Since the usual trace pairing is nondegenerate, we have (cid:80) r − j =1 (cid:80) j − l =0 c ( i ) j ( t N ( i ) ) l ◦ ξ ( i ) ◦ ( N ( i ) ) j − − l = 0. Recall that Θ ( i )( θ ( i ) ◦ ξ ( i ) ) = 0 by the choice of ( ξ ( i ) ), whichis equivalent to the existence of some g ∈ End( E | p ) satisfying θ ( i ) ◦ ξ ( i ) = N ( i ) ◦ g − g ◦ N ( i ) . So we have (cid:80) r − j =1 c ( i ) j ( θ ( i ) ) − ◦ (( N ( i ) ) j ◦ g − g ◦ ( N ( i ) ) j ) = 0, which means ν ( i ) ( N ( i ) ) ◦ g = g ◦ ν ( i ) ( N ( i ) ). Since ν ( i ) satisfies Assumption 2.2, we have N ( i ) ◦ g = g ◦ N ( i ) and ξ ( i ) = 0. Thus the pairing (28) is nondegeneratebecause rank O D ker (cid:0) S ( E | D s , E | ∨D s ) → Z (cid:1) = r ( r − O D coker (cid:0) Z → S ( E | ∨D s , E | D s ) (cid:1) . So the pair-ing (27) becomes a nondegenerate pairing of vector spaces over C and η becomes isomorphic. Thus thehomomorphism σ : H ( F • ) ∼ −→ H ( F • ) becomes an isomorphism by the five lemma. The homomorphism σ : H ( F • ) ∼ −→ H ( F • ) is isomorphic because it is the dual of σ .On the other hand, we have the exact commutative diagramker( Z → S ( E | ∨D s , E | D s )) −→ H ( F • ) −→ H ( G ) −→ coker( Z → S ( E | ∨D s , E | D s )) (cid:121) σ (cid:121) η (cid:121) t η (cid:121) ∼ = coker( S ( E | D s , E | ∨D s ) → Z ) ∨ −→ H ( F • ) ∨ −→ H (ker( G → G )) ∨ −→ ker( S ( E | D s , E | ∨D s ) → Z ) ∨ . Note that ker( Z → S ( E | ∨D s , E | D s )) = 0 and coker( S ( E | D s , E | ∨D s ) → Z ) = 0. The homomorphism η isisomorphic since it is induced by the isomorphism ker( G → G ) ∨ ⊗ Ω C s ∼ = G and the Serre duality. Thusthe homomorphism σ is an isomorphism. The homomorphism σ : H ( F • ) −→ H ( F • ) ∨ is isomorphic,because it is the dual of σ .From all the above arguments, the homomorphism σ : H ( F • ) −→ H ( F • ) ∨ is isomorphic by the fivelemma, because σ , σ , σ , σ are all isomorphic. (cid:3) Lemma 2.21. H (Tr) : H ( F • ) −→ H ( ˜ L • s ) ∼ = C is an isomorphism.Proof. From the proof of Lemma 2.20, the exact commutative diagram H ( F • ) −−−−→ H ( F • ) −−−−→ H ( F • ) −−−−→ σ (cid:121) σ (cid:121) σ (cid:121) H ( F • ) ∨ −−−−→ H ( F • ) ∨ −−−−→ H ( F • ) ∨ −−−−→ σ : H ( F • ) ∼ −→ H ( F • ) ∨ is an isomorphism because σ and σ are isomorphic. Note that H ( F • ) = C because ( E, ∇ , { N ( i ) } ) is α -stable whose endomorphisms are only scalar multiplications. Wecan see from the construction that the composition H ( F • ) σ −→ ∼ H ( F • ) ∨ ∼ −→ H ( ˜ L • s ) ∨ ∼ −→ H ( ˜ L • s )coincides with H (Tr) and the result follows. (cid:3) Corollary 2.22. The dimension of the relative tangent space of M α C , D (˜ ν , ˜ µ ) over S at ( E, ∇ , { N ( i ) } ) isgiven by dim H ( F • ) = 2 r ( g − 1) + 2 + r ( r − n (cid:88) i =1 m i . Proof. Since we will prove the smoothness of the moduli space M α C , D (˜ ν , ˜ µ ) over S in Proposition 2.25, wecan deduce the corollary from [17, Theorem 2.1] and [19, Theorem 2.2]. We give here a direct proof usingthe proof of Lemma 2.20. Since H ( F • ) ∼ = C and H ( F • ) ∼ = C , the exact sequence (20) becomes0 −→ C −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ H ( F • ) −→ C −→ . Since H ( F • ) ∼ = H ( F • ) ∨ and H ( F • ) ∼ = H ( F • ) ∨ by the proof of Lemma 2.20, we havedim H ( F • ) = dim H ( F • ) + dim H ( F • ) − dim H ( F • ) − dim H ( F • ) + dim C + dim C (29) = 2 dim H ( F • ) − H ( F • ) + 2= − χ ( F • ) + 2Using the Riemann-Roch formula, we can see χ ( F • ) = χ ( G ) + length Z − length S ( E | ∨D s , E | D s ))= r (1 − g ) + n (cid:88) i =1 rm i − n (cid:88) i =1 r ( r + 1)2 m i . Substituting this in (29) we get the corollary. (cid:3) Smoothness of the moduli space of (˜ ν , ˜ µ ) -connections. We use the same notations as in sub-section 2.4 and subsection 2.5. Proposition 2.23. Let A be an artinian local ring over S with the maximal ideal m and I be an ideal of A satisfying m I = 0 and A/ m = C . Let ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) j } ) be a flat family of (˜ ν , ˜ µ ) ⊗ A/I -connections on ( C A/I , D A/I ) over A/I such that ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ⊗ A/ m ∼ = ( E, ∇ , { N ( i ) } ) . Then there is an obstruction class o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) ∈ H ( F • ) ⊗ I whose vanishing is equivalent to the existence of a lift of ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) j } ) to a flat family of (˜ ν , ˜ µ ) ⊗ A -connections on ( C A , D A ) over A .Proof. We can define the O D ( i ) A/I [ T ]-module structures on E (cid:48) | D ( i ) A/I and on E (cid:48)∨ | D ( i ) A/I by N (cid:48) ( i ) and t N (cid:48) ( i ) ,respectively. Then we can take an O D ( i ) A/I [ T ]-isomorphism θ (cid:48) ( i ) : E (cid:48)∨ | D ( i ) A/I ∼ −→ E (cid:48) | D ( i ) A/I which is a lift of NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 27 θ ( i ) . If we put κ (cid:48) ( i ) := ( θ (cid:48) ( i ) ) − ◦ N (cid:48) ( i ) : E (cid:48) | D ( i ) A/I −→ E (cid:48)∨ | D ( i ) A/I , then ( E (cid:48) , ∇ (cid:48) , { θ (cid:48) ( i ) , κ (cid:48) ( i ) } ) is a flat family offactorized (˜ ν , ˜ µ ) ⊗ A/I -connections on ( C A/I , D A/I ) over A/I .We can take an affine open covering C A = (cid:83) α U α such that (cid:93) { i | D ( i ) A ∩ U α (cid:54) = ∅} ≤ α and (cid:93) { α | D ( i ) A ⊂ U α } = 1 for any i . Furthermore, we may assume that E (cid:48) | U α ⊗ A/I ∼ = O ⊕ rU α ⊗ A/I . Take a free O U α -module E α with an isomorphism ψ α : E α ⊗ A/I ∼ −→ E (cid:48) | U α ⊗ A/I and a lift σ βα : E α | U αβ ∼ −→ E β | U αβ ofthe composite ψ − β ◦ ψ α : E α | U αβ ⊗ A/I ψ α −−→ ∼ E (cid:48) | U αβ ⊗ A/I ψ − β −−−→ ∼ E β | U αβ ⊗ A/I .If we write ϕ ( i )˜ µ ⊗ A ( T ) = T r + b r − T r − + · · · + b T + b with b i ∈ O D ( i ) A and define matrices N, Φ , Φ by N = − b r − · · · − b r − · · · ... ... . . . . . . ... − b · · · − b · · · · · · , Φ = · · · · · · b r − ... ... . . . . . . ... b r − · · · b b r − b r − · · · b , Φ = · · · · · · b r − ... ... . . . . . . ... ... b r − · · · b b r − b r − · · · b 00 0 0 · · · − b , then t Φ = Φ , t Φ = Φ and Φ is invertible. We can check N Φ = Φ , which is equivalent to N = Φ Φ − .So there is a matrix factorization t N = Φ − Φ : O ⊕ r D ( i ) A Φ −−→ (cid:16) O ⊕ r D ( i ) A (cid:17) ∨ Φ − −−−→ O ⊕ r D ( i ) A . After replacing the representative (( θ (cid:48) ( i ) ) , ( κ (cid:48) ( i ) )) by the action of an element of (cid:18) O D ( i ) A/I [ T ] / ( ϕ ( i )˜ µ ⊗ A/I ( T )) (cid:19) × ,we may assume that there is an isomorphism g : O ⊕ r D ( i ) A/I ∼ −→ E (cid:48) | D ( i ) A/I satisfying θ (cid:48) ( i ) = g ◦ (Φ − ⊗ A/I ) ◦ t g and κ (cid:48) ( i ) = t g − ◦ (Φ ⊗ A/I ) ◦ g − . We take a lift ˜ g : O ⊕ r D ( i ) A ∼ −→ E α | D ( i ) A of g , that is, ψ α ◦ (˜ g ⊗ A/I ) = g .If we put θ ( i ) α := ˜ g ◦ Φ − ◦ t ˜ g and κ ( i ) α := ( t ˜ g ) − ◦ Φ ◦ ˜ g − , then ( θ ( i ) α , κ ( i ) α ) becomes a lift of ( θ (cid:48) ( i ) , κ (cid:48) ( i ) )and N ( i ) α := θ ( i ) α ◦ κ ( i ) α : E α | D ( i ) A −→ E α | D ( i ) A becomes a lift of N (cid:48) ( i ) . We can take an A -relative localconnection ∇ α : E α −→ E α ⊗ Ω C A /A ( D A ) satisfying ν ( i ) ( N ( i ) α ) dz ( i ) ¯ z ( i )1 ¯ z ( i )2 · · · ¯ z ( i ) m i = ∇ α | D ( i ) A and ∇ α ⊗ A/I = ψ − α ◦ ∇ (cid:48) | U α ⊗ A/I ◦ ψ α .If we put u αβγ = ψ α ◦ ( σ − γα ◦ σ γβ ◦ σ βα − id E α ) ◦ ψ − α , v αβ = ψ α ◦ ( σ − βα ◦ ∇ β ◦ σ βα − ∇ α ) ◦ ψ − α , then we have v βγ − v αγ + v αβ = ∇ (cid:48) ◦ u αβγ − u αβγ ◦ ∇ (cid:48) , u βγδ − u αγδ + u αβδ − u αβγ = 0and we can define an element o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) := [( { ( u αβγ , } , { ( v αβ , (0 , } , { (0 , } )] ∈ H ( F • ) ⊗ I. Assume that o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) = 0. Then there are { a αβ } ∈ I ⊗ C ( { U α } , G ) , { b α , ( τ ( i ) α ) , ( ξ ( i ) α ) } ∈ I ⊗ C ( { U α } , G ⊕ S ( E ∨ | D s , E | D s ) ⊕ S ( E | D s , E | ∨D s ))satisfying u αβγ = a βγ − a αγ + a αβ , v αβ = ∇ a αβ − a αβ ∇ − ( b β − b α ) ,b α | D ( i ) s = δ ( i ) ν ,N ( i ) ( τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α ) , Θ ( i ) τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α = 0 . If we put ˜ θ ( i ) α := θ ( i ) α + ψ − α ◦ τ ( i ) α ◦ ψ α , ˜ κ ( i ) α := κ ( i ) α + ψ − α ◦ ξ ( i ) α ◦ ψ α , then the composition ˜ N ( i ) α := ˜ θ ( i ) α ◦ ˜ κ ( i ) α = N ( i ) α + ψ − α ◦ ( τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α ) ◦ ψ α satisfies ϕ ( i )˜ µ ( ˜ N ( i ) α ) = 0, because there is g ( i ) α ∈ End( E | D ( i ) s ) ⊗ I satisfying N ( i ) ◦ g ( i ) α − g ( i ) α ◦ N ( i ) = τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α from the condition Θ ( i ) τ ( i ) α ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) α = 0. We definea connection ˜ ∇ α on E α by ˜ ∇ α := ∇ α + ψ − α ◦ b α ◦ ψ α . Then we have˜ ∇ α | D ( i ) A = ∇ α | D ( i ) A + ( ψ − α ◦ b α ◦ ψ α ) | D ( i ) A = ˜ ν ( i ) ( N ( i ) α ) d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i + δ ( i ) ν ,N ( i ) ( ˜ N ( i ) α − N ( i ) α )= ˜ ν ( i ) ( N ( i ) α ) + r − (cid:88) j =1 j (cid:88) l =1 c ( i ) j ( ˜ N ( i ) α ) j − l ( ˜ N ( i ) α − N ( i ) α )( N ( i ) α ) l − d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = ˜ ν ( i ) ( N ( i ) α ) + r − (cid:88) j =0 c ( i ) j ( ˜ N ( i ) α ) j d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i − r − (cid:88) j =0 c ( i ) j ( N ( i ) α ) j d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i = ˜ ν ( i ) ( ˜ N ( i ) α ) d ¯ z ( i ) ¯ z ( i )1 · · · ¯ z ( i ) m i . If we put ˜ σ βα := σ βα ◦ (id − ψ − α ◦ a αβ ◦ ψ α ), then(˜ σ γα ) − ◦ ˜ σ γβ ◦ ˜ σ βα = (id + ψ − α ◦ a αγ ◦ ψ α ) ◦ σ − γα ◦ σ γβ ◦ (id − ψ − β ◦ a βγ ◦ ψ β ) ◦ σ βα ◦ (id − ψ − α ◦ a αβ ◦ ψ α )= (id + ψ − α ◦ a αγ ◦ ψ α ) ◦ σ − γα ◦ σ γβ ◦ σ βα ◦ (id − ψ − α ◦ a βγ ◦ ψ α ) ◦ (id + ψ − α ◦ a αβ ◦ ψ α )= σ − γα ◦ σ γβ ◦ σ βα ◦ (id + ψ − α ◦ a αγ ◦ ψ α ) ◦ (id − ψ − α ◦ a βγ ◦ ψ α ) ◦ (id − ψ − α ◦ a αβ ◦ ψ α )= (id + ψ − α ◦ u αβγ ◦ ψ α ) ◦ (id + ψ − α ◦ ( a αγ − a βγ − a αβ ) ◦ ψ α )= id + ψ − α ◦ ( u αβγ − ( a βγ − a αγ + a αβ )) ◦ ψ α = idbecause σ βα ⊗ A/I = id. We also have˜ σ − βα ◦ ˜ ∇ β ◦ ˜ σ βα = (id + ψ − α ◦ a αβ ◦ ψ α ) ◦ σ − βα ◦ ( ∇ β + ψ − β ◦ b β ◦ ψ β ) ◦ σ βα ◦ (id − ψ − α ◦ a αβ ◦ ψ α )= σ − βα ◦ ∇ β ◦ σ βα − ψ − α ◦ ∇ (cid:48) ◦ a αβ ◦ ψ α + ψ − α ◦ a αβ ◦ ∇ (cid:48) ◦ ψ α + ψ − α ◦ b β ◦ ψ α = ∇ α + ψ − α ◦ v αβ ◦ ψ α − ψ − α ◦ ( ∇ (cid:48) ◦ a αβ − a αβ ◦ ∇ (cid:48) − b β ) ◦ ψ α = ∇ α + ψ − α ◦ b α ◦ ψ α = ˜ ∇ α . Thus we can patch ( E α , ˜ ∇ α , { ˜ θ ( i ) α , ˜ κ ( i ) α } ) together via the gluing isomorphisms { ˜ σ βα } and obtain a flatfamily ( ˜ E, ˜ ∇ , { ˜ θ ( i ) , ˜ κ ( i ) } ) of factorized (˜ ν , ˜ µ ) ⊗ A -connections over A which is a lift of ( E (cid:48) , ∇ (cid:48) , { θ (cid:48) ( i ) , κ (cid:48) ( i ) } ).Conversely, we can immediately see that o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) = 0 if there is a lift of ( E (cid:48) , ∇ (cid:48) , { θ (cid:48) ( i ) , κ (cid:48) ( i ) } ) over A , which corresponds to a lift of ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) over A . Thus the proposition is proved. (cid:3) Lemma 2.24. The isomorphism H (Tr) : H ( F • ⊗ I ) ∼ −→ H ( ˜ L • s ⊗ I ) = H ( L • s ⊗ I ) in Lemma 2.21 sendsthe obstruction class o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) } ) defined in the proof of Proposition 2.23 to an element of H ( L • s ⊗ I ) whose vanishing is equivalent to the existence of an extension of (det( E (cid:48) , ∇ (cid:48) )) to a pair ( L, ∇ L ) of a linebundle L on C × Spec A and a connection ∇ L : L −→ L ⊗ Ω C A /A ( D A ) satisfying ( L, ∇ L ) ⊗ A/I ∼ = det( E (cid:48) , ∇ (cid:48) ) and ∇ L | D ( i ) A = (cid:80) rk =1 ˜ ν ( i ) (˜ µ ( i ) k ) A .Proof. Take the same affine open covering { U α } of C A and the lifts ( E α , ∇ α ) of ( E (cid:48) , ∇ (cid:48) ) | U α × Spec A/I as inthe proof of Proposition 2.23. Then det( E α , ∇ α ) is a lift of det( E (cid:48) , ∇ (cid:48) ) | U α × Spec A/I and the class o (det( E (cid:48) , ∇ (cid:48) )) := (cid:104)(cid:8) det( ψ α ) ◦ (det( σ − γα ◦ σ γβ ◦ σ βα ) − id det E α ) ◦ det( ψ − α ) (cid:9) , (cid:8) det( ψ α ) ◦ (det( σ − βα ) ◦ det( ∇ β ) ◦ det( σ βα ) − det( ∇ α )) ◦ det( ψ − α ) (cid:9)(cid:105) ∈ H ( L • ⊗ I )is nothing but the obstruction for the existence of a lift ( L, ∇ L ) of det( E (cid:48) , ∇ (cid:48) ) over A satisfying ∇ L | D ( i ) A = (cid:80) rk =1 ˜ ν ( i ) (˜ µ ( i ) k ) A . Here det ∇ α : det E α −→ det E α ⊗ Ω C A /A ( D A ) is the A -relative connection on det( E α )induced from ∇ α , which is defined by(det( ∇ α ))( v ∧ v ∧ · · · v r ) = ∇ α ( v ) ∧ v ∧ · · · ∧ v r + · · · + v ∧ · · · ∧ v r − ∧ ∇ α ( v r ) NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 29 for v , . . . , v r ∈ E α . For the notations { u αβγ } , { v αβ } in the proof of Proposition 2.23, we haveTr( u αβγ ) = det( ψ α ) ◦ (det( σ − γα ◦ σ γβ ◦ σ βα ) − id det E α ) ◦ det( ψ − α )Tr( v αβ ) = det( ψ α ) ◦ (det( σ − βα ) ◦ det( ∇ β ) ◦ det( σ βα ) − det( ∇ α )) ◦ det( ψ − α ) . So o (det( E (cid:48) , ∇ (cid:48) )) is nothing but the image of the obstruction class o ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) j , N (cid:48) ( i ) j } ) ∈ H ( F • ⊗ I )under the isomorphism H (Tr) : H ( F • ⊗ I ) ∼ −→ H ( L • s ⊗ I ). (cid:3) Proposition 2.25. The moduli space M α C , D (˜ ν , ˜ µ ) is smooth over S .Proof. Consider the S -relative moduli space M C , D (Tr(˜ ν ) , Tr( ˜ µ )) whose S (cid:48) -valued points are the pairs( L, ∇ L ) of a line bundle L on C S (cid:48) and a relative connection ∇ L : L −→ L ⊗ Ω C S (cid:48) /S (cid:48) ( D S (cid:48) ) satisfying ∇ L | D ( i ) S (cid:48) = (cid:80) rk =1 ˜ ν ( i ) (˜ µ ( i ) k ) S (cid:48) . Then M C , D (Tr(˜ ν ) , Tr( ˜ µ )) is an affine space bundle over the Jacobian va-riety of C over S whose fiber is isomorphic to H (Ω C s ). So we can prove by the same method as in the proofof [17, Theorem 2.1] that M C , D (Tr(˜ ν ) , Tr( ˜ µ )) is smooth over S and the obstruction class o (det( E (cid:48) , ∇ (cid:48) ))should vanish. Thus the obstruction class o ( E (cid:48) , ∇ (cid:48) , { N (cid:48) ( i ) j } ) also vanishes by Lemma 2.24 and the modulispace M α C , D (˜ ν , ˜ µ ) is smooth over S . (cid:3) Relative symplectic form on the moduli space.Proposition 2.26. There exists an S -relative symplectic form ω ∈ H ( M α C , D (˜ ν , ˜ µ ) , Ω M α C , D (˜ ν , ˜ µ ) /S ) on themoduli space M α C , D (˜ ν , ˜ µ ) .Proof. For some quasi-finite ´etale covering ˜ M −→ M α C , D (˜ ν , ˜ µ ), there is a universal flat family of (˜ ν , ˜ µ )-connections ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) on C × S ˜ M over ˜ M . Replacing ˜ M by a refinement, there is a corresponding flatfamily ( ˜ E, ˜ ∇ , { ˜ θ ( i ) , ˜ κ ( i ) } ) of factorized (˜ ν , ˜ µ )-connections on C × S ˜ M over ˜ M . We define homomorphisms σ ( i ) − θ ( i ) : E nd ( E | D ( i )˜ M ) ⊕ O ˜ M [ T ] / ( ϕ ( i )˜ µ ( T )) −→ H om ( E | ∨D ( i )˜ M , E | D ( i )˜ M ) σ ( i )+ κ ( i ) : E nd ( E | D ( i )˜ M ) ⊕ O ˜ M [ T ] / ( ϕ ( i )˜ µ ( T )) −→ H om ( E | D ( i )˜ M , E | ∨D ( i )˜ M ) δ ( i ) ν ,N ( i ) : E nd ( E | D ( i )˜ M ) −→ E nd ( E | D ( i )˜ M ) ⊗ Ω C ˜ M / ˜ M ( D ˜ M )by the same formulas as in subsection 2.4, (15), (16) and (17). For each u ∈ E nd ( E | D ( i )˜ M ), we define ahomomorphism Θ ( i ) u : O D ( i )˜ M [ T ] / ( ϕ ( i )˜ µ ( T )) −→ Ω C ˜ M / ˜ M ( D ˜ M ) | D ( i )˜ M by the same formula as subsection 2.4, (18). We put˜ G := E nd ( ˜ E ) , ˜ G := E nd ( ˜ E ) ⊗ Ω C× S ˜ M/ ˜ M ( D ˜ M ) , ˜ G := ˜ G | D ˜ M ,S ( ˜ E | ∨D ˜ M , ˜ E | D ˜ M ) := (cid:40) ( τ ( i ) ) ∈ n (cid:77) i =1 H om ( ˜ E | ∨D ( i )˜ M , ˜ E | D ( i )˜ M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t τ ( i ) = τ ( i ) for any i (cid:41) ,S ( ˜ E | D ˜ M , ˜ E | ∨D ˜ M ) := (cid:40) ( ξ ( i ) ) ∈ n (cid:77) i =1 H om ( ˜ E | D ( i )˜ M , ˜ E | ∨D ( i )˜ M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ξ ( i ) = ξ ( i ) for any i (cid:41) , ˜ Z := n (cid:77) i =1 O D ( i )˜ M [ T ] / ( ϕ ( i )˜ µ ( T )) , ˜ Z := n (cid:77) i =1 H om O D ( i )˜ M ( O D ( i )˜ M [ T ] / ( ϕ ( i )˜ µ ( T )) , Ω C ˜ M / ˜ M ( D ˜ M ) (cid:12)(cid:12) D ˜ M ) . We define a complex ˜ F • = [ ˜ F d −→ ˜ F d −→ ˜ F ] in the same way as subsection 2.4;˜ F = ˜ G ⊕ ˜ Z , ˜ F = ˜ G ⊕ S ( ˜ E | ∨D ˜ M , ˜ E | D ˜ M ) ⊕ S ( ˜ E | D ˜ M , ˜ E | ∨D ˜ M ) , ˜ F = ˜ G ⊕ ˜ Z d ( u, ( P ( i ) ( T ))) = (cid:16) ∇ ◦ u − u ◦ ∇ , (cid:16) σ ( i ) − θ ( i ) (cid:16) u | D ( i ) s , P ( i ) ( T ) (cid:17)(cid:17) , (cid:16) σ ( i )+ κ ( i ) (cid:16) u | D ( i ) s , P ( i ) ( T ) (cid:17)(cid:17)(cid:17) d ( v, ( τ ( i ) ) , ( ξ ( i ) )) = (cid:16)(cid:16) v | D ( i ) s − δ ( i ) ν ,N ( i ) ( τ ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) ) (cid:17) , (cid:16) Θ ( i )( τ ( i ) ◦ κ ( i ) + θ ( i ) ◦ ξ ( i ) ) (cid:17)(cid:17) . Then we can see by the same proof as Proposition 2.17 that the relative tangent bundle T ˜ M/S of ˜ M over S is isomorphic to R ( p ˜ M ) ∗ ( ˜ F • ), where p ˜ M : C × S ˜ M −→ ˜ M is the structure morphism. We define (cid:0) Ξ ( τ ( i ) ,ξ ( i ) )( τ (cid:48) ( i ) ,ξ (cid:48) ( i ) ) (cid:1) ∈ Ω C ˜ M / ˜ M ( D ˜ M ) | D ˜ M for (( τ ( i ) ) , ( ξ ( i ) )) , (( τ (cid:48) ( i ) ) , ( ξ (cid:48) ( i ) )) ∈ S ( ˜ E | ∨D ˜ M , ˜ E | D ˜ M ) ⊕ S ( ˜ E | D ˜ M , ˜ E | ∨D ˜ M ) inthe same way as (21) in subsection 2.5. We take an affine open covering { U α } of C and define a pairing ω ˜ M : R ( p ˜ M ) ∗ ( ˜ F • ) × R ( p ˜ M ) ∗ ( ˜ F • ) −→ R ( p ˜ M ) ∗ ( L • M ) ∼ = O ˜ M by ω ˜ M (cid:16)(cid:2)(cid:8) ( u αβ , (cid:9) , (cid:8) ( v α , (( τ ( i ) α ) , ( ξ ( i ) α ))) (cid:9)(cid:3) , (cid:2)(cid:8) ( u (cid:48) αβ , (cid:9) , (cid:8) ( v (cid:48) α , (( τ (cid:48) ( i ) α ) , ( ξ (cid:48) ( i ) α ))) (cid:9)(cid:3)(cid:17) = (cid:104)(cid:8) Tr( u αβ ◦ u (cid:48) βγ ) (cid:9) , − (cid:8)(cid:0) Tr( u αβ ◦ v (cid:48) β − v α ◦ u (cid:48) αβ ) , (cid:1)(cid:9) , (cid:110)(cid:0) Ξ ( τ ( i ) α ,ξ ( i ) α )( τ (cid:48) ( i ) α ,ξ (cid:48) ( i ) α ) (cid:1)(cid:111)(cid:105) using the ˘Cech cohomology with respect to the covering { U α × S ˜ M } . Then the restriction ω ˜ M (cid:12)(cid:12) x at a point x of ˜ M whose image in M α C , D (˜ ν , ˜ µ ) corresponds to ( E, ∇ , { l ( i ) } ) is nothing but the pairing ω ( E, ∇ , { l ( i ) } ) inLemma 2.20, which is nondegenerate. We can easily see that ω ˜ M descends to a pairing ω M α C , D (˜ ν , ˜ µ ) : T M α C , D (˜ ν , ˜ µ ) /S × T M α C , D (˜ ν , ˜ µ ) /S −→ O M α C , D (˜ ν , ˜ µ ) which is nondegenerate. If we take a tangent vector v ∈ T M α C , D (˜ ν , ˜ µ ) /S ( x ) at a point x ∈ M α C , D (˜ ν , ˜ µ ) cor-responding to a (˜ ν s , ˜ µ s )-connection ( E, ∇ , { l ( i ) } ), v corresponds to a C [ t ] / ( t )-valued point ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) } )of M α C , D (˜ ν , ˜ µ ) which is a lift of ( E, ∇ , { l ( i ) } ). Then we can check that ω M α C , D (˜ ν , ˜ µ ) ( v, v ) coincides with theimage by Tr : H ( F • ) ∼ −→ H ( L • s ) of the obstruction class o ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) } ) for the lifting of ( E (cid:48) , ∇ (cid:48) , { l (cid:48) ( i ) } )to a C [ t ] / ( t )-valued point of M α C , D (˜ ν , ˜ µ ) which is given in Proposition 2.23. Since M α C , D (˜ ν , ˜ µ ) is smoothover S by Proposition 2.25, we have ω M α C , D (˜ ν , ˜ µ ) ( v, v ) = 0. So the pairing ω M α C , D (˜ ν , ˜ µ ) is skew-symmetric anddefine a relative 2-form ω M α C , D ∈ H ( M α C , D (˜ ν , ˜ µ ) , Ω M α C , D (˜ ν , ˜ µ ) /S ).A generic geometric fiber M α C , D (˜ ν , ˜ µ ) s over S is the moduli space of regular singular connections on C s along the reduced divisor D s . If we put ˜ M s := ˜ M × M α C , D (˜ ν , ˜ µ ) M α C , D (˜ ν , ˜ µ ) s , there is a universal parabolicstructure ˜ E ˜ M s | ( ˜ D ( i ) j ) ˜ Ms = ˜ l ( i ) j, ⊃ · · · ⊃ ˜ l ( i ) j,r − ⊃ ˜ l ( i ) j,r = 0 determined by ˜ ∇ ˜ M s . If we put˜ F par := (cid:110) u ∈ ˜ G M α C , D (˜ ν , ˜ µ ) s (cid:12)(cid:12)(cid:12) u | ( D ( i ) j ) ˜ Ms (˜ l ( i ) j,k ) ⊂ ˜ l ( i ) j,k for any i, j, k (cid:111) ˜ F par := (cid:110) v ∈ ˜ G M α C , D (˜ ν , ˜ µ ) s (cid:12)(cid:12)(cid:12) v | ( D ( i ) j ) ˜ Ms (˜ l ( i ) j,k ) ⊂ ˜ l ( i ) j,k +1 ⊗ Ω C ˜ Ms / ˜ M s ( D ˜ M s ) for any i, j, k (cid:111) ∇ ˜ F • par : ˜ F par (cid:51) u (cid:55)→ ˜ ∇ ◦ u − u ◦ ˜ ∇ ∈ ˜ F par , then the canonical inclusions ˜ F par (cid:44) → ˜ G M s and ˜ F par (cid:44) → ˜ G M s induce a morphism ˜ F • par −→ ˜ F • ˜ M s of complexeswhich induces an isomorphism R ( π ˜ M s ) ∗ ( ˜ F • par ) ∼ −→ R ( π ˜ M s ) ∗ ( ˜ F • ˜ M s )because they are both isomorphic to the tangent bundle of ˜ M s . A symplectic form ω ˜ M s on ˜ M s is definedin [17, Proposition 7.2], which satisfies dω ˜ M s = 0 by [17, Porposition 7.3]. By construction, we can see that ω ˜ M s = ω ˜ M | ˜ M s . So we have dω M α C , D (˜ ν , ˜ µ ) | M α C , D (˜ ν , ˜ µ ) s = 0, which implies that ω M α C , D (˜ ν , ˜ µ ) is relatively d -closedon M α C , D (˜ ν , ˜ µ ) over S . (cid:3) Eventually Theorem 2.11 follows from Corollary 2.22, Proposition 2.25 and Proposition 2.26. .3. Fundamental solution of an unfolded linear differential equation with an asymptoticproperty In this section, we introduce the existence theorem of fundamental solutions with an asymptotic propertyof an unfolded linear differential equation, which is one of the main tools in the unfolding theory of lineardifferential equations established by Hurtubise, Lambert and Rousseau in [14] and [15]. Unfortunately, theunfolded generalized isomonodromic deformation in Theorem 0.1 is not compatible with the asymptoticproperty given in the unfolding theory in [14], [15]. However, it will be worth pointing out what is thedifficulty in adopting the asymptotic property in [14], [15] to our moduli theoretic setting constructed insection 2. Since the unfolding theory in [14], [15] are written in a very general setting and hard to followall of them, we restrict to the easy case when the unfolding of the singular divisor is given by the equation z m − (cid:15) m = 0. NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 31 Flows for an asymptotic estimate. Let ∆ = { z ∈ C | | z | < } be a unit disk in the complexplane C . For an integer m with m ≥ 2, we put ζ m := exp (cid:18) π √− m (cid:19) . Then we have z m − (cid:15) m =( z − (cid:15)ζ m )( z − (cid:15)ζ m ) · · · ( z − (cid:15)ζ mm ) for z, (cid:15) ∈ ∆. We set D := { ( z, (cid:15) ) ∈ ∆ × ∆ | z m − (cid:15) m = 0 } . Note that there is an equality1 z m − (cid:15) m = 1( z − (cid:15)ζ m ) · · · ( z − (cid:15)ζ mm ) = m (cid:88) j =1 (cid:81) j (cid:54) = i (cid:15) ( ζ im − ζ jm ) 1 z − (cid:15)ζ im for ( z, (cid:15) ) ∈ (∆ × ∆) \ D . By Lemma 2.1, we have m (cid:88) i =1 (cid:81) j (cid:54) = i ( (cid:15)ζ im − (cid:15)ζ jm ) = m (cid:88) i =1 res z = (cid:15)ζ im (cid:18) dz ( z − (cid:15)ζ m )( z − (cid:15)ζ m ) · · · ( z − (cid:15)ζ mm ) (cid:19) = 0for (cid:15) (cid:54) = 0, since m ≥ θ ∈ R , we consider a holomorphic differential equation(30) dzdτ = e √− θ ( z m − (cid:15) m ) = e √− θ ( z − (cid:15)ζ m )( z − (cid:15)ζ m ) · · · ( z − (cid:15)ζ mm ) . Under the above equation, we can regard τ as a multi-valued function in z ∈ (∆ × ∆) \ D . We substituteinto τ ∈ C a real variable t ∈ R and consider the restricted differential equation(31) dzdt = e √− θ ( z m − (cid:15) m ) = e √− θ ( z − (cid:15)ζ m )( z − (cid:15)ζ m ) · · · ( z − (cid:15)ζ mm ) . Note that giving a solution z ( t ) = x ( t ) + √− y ( t ) of the differential equation (31) is equivalent to giving aflow of the vector field(32) v (cid:15),θ = Re (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂x + Im (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂y . For the investigation of the flow of the vector field v (cid:15),θ , we consider the surjective morphism (cid:36) : ∆ × [0 , × S −→ ∆ × ∆defined by (cid:36) ( z, s, e √− ψ ) = ( z, se √− ψ )and we call (cid:36) a polar blow up of ∆ × ∆ along ∆ × { } . Here we denote { t ∈ R | a ≤ t < b } by [ a, b ) for realnumbers a, b satisfying a < b .We consider the following proposition which treats an easy restricted case of the analysis of flows in aseries of papers [29], [30], [14], [15]. We give here just an elementary proof in an easy restricted case for thepurpose of the author’s understanding. So it may seem trivial for experts. Proposition 3.1. There is an open neighborhood U of { } × { } × S in ∆ × [0 , × S and an opencovering (33) U \ ( U ∩ (cid:36) − ( D )) = m (cid:91) j =1 (cid:91) ≤ ψ ≤ π (cid:91) ξ =1 , W ( j ) ψ ,ξ such that any flow of the vector field v (cid:15),θ ( j ) ψ ,ξ = Re (cid:16) e √− θ ( j ) ψ ,ξ ( z m − (cid:15) m ) (cid:17) ∂∂x + Im (cid:16) e √− θ ( j ) ψ ,ξ ( z m − (cid:15) m ) (cid:17) ∂∂y starting at a point of W ( j ) ψ ,ξ converges to a point in (cid:36) − ( D ) , where θ ( j ) ψ ,ξ is determined by j, ψ , ξ .Proof. We take a point ( z , s , e √− ψ ) ∈ (∆ \ { } ) × (cid:2) , (cid:1) × S satisfying 0 < | z | < . We can choosean integer j with 1 ≤ j ≤ m satisfying − πm ≤ arg( z ) − ψ − jπm ≤ πm . We divide into two cases:0 ≤ arg( z ) − ψ − jπm ≤ πm , − πm ≤ arg( z ) − ψ − jπm < . Case 1. ≤ arg( z ) − jπm − ψ ≤ πm .In this case we choose small δ > δ < π m and put(34) θ ( j ) ψ , := − j ( m − πm − ( m − ψ + π + δ. We simply denote θ ( j ) ψ , by θ in the following. So θ is given by θ − πm − − jπm − ψ + δm − . Note that we have δm − ≤ arg( z ) + θ − πm − ≤ πm + δm − , ψ + 2 jπm + θ − πm − δm − . If we replace δ > l = (cid:26) z ∈ C (cid:12)(cid:12)(cid:12)(cid:12) arg (cid:16) e √− π m − e √− θ − πm − z (cid:17) = (2 m + 1) δm − , | z | < , Re( z ) > (cid:27) l = (cid:26) z ∈ C (cid:12)(cid:12)(cid:12)(cid:12) arg( z ) + θ − πm − πm + 2 δm − , | z | < , Re( z ) > (cid:27) intersects at a point s e √− πm − θ − πm − + δm − ) satisfying 14 < s < 1. Then we put P ( j ) ψ , = ( z, ( s, e √− ψ )) ∈ ∆ × (cid:104) , (cid:17) × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − δ m − < ψ + 2 jπm + θ − πm − < δ m − , z (cid:54) = 0(2 m + 1) δm − < arg (cid:16) e √− π m − e √− θ − πm − z (cid:17) < π π m and − π m < arg( z ) + θ − πm − < πm + 2 δm − . A picture of the region (cid:110) ˜ z = e √− θ − πm − z (cid:12)(cid:12)(cid:12) ( z, s, e √− ψ ) ∈ P ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17)(cid:111) looks like [figure 1].Since arg (cid:16) e √− θ − πm − e √− θ (cid:0) e √− ψ ζ jm (cid:1) m (cid:17) = θ − πm − θ + mψ = m ( θ − π ) m − mψ + π , we have(35) π − mδ m − < arg (cid:16) e √− θ − πm − e √− θ (cid:0) e √− ψ ζ jm (cid:1) m (cid:17) < π + 3 mδ m − − δ m − < ψ + 2 jπm + θ − πm − < δ m − η > m, j, θ, δ such that(36) − mδm − < arg (cid:16) e √− θ − πm − e √− θ ( w m − ( e √− ψ ζ jm ) m ) (cid:17) < mδm − w ∈ ∆ satisfying | w | ≤ η , when − δ m − < ψ + 2 jπm + θ − πm − < δ m − Q ( j ) ψ , := ( z, s, e √− ψ ) ∈ ∆ × (cid:104) , (cid:17) × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e √− θ − πm − z − ηse √− π (cid:54) = 0 , − π m < arg (cid:16) e √− θ − πm − z − ηse √− π (cid:17) < π m − δ m − < ψ + 2 jπm + θ − πm − < δ m − πm + 2 δm − ≤ arg( z ) + θ − πm − ≤ π − π m if z (cid:54) = 0 . and set R ( j ) ψ , := P ( j ) ψ , ∪ Q ( j ) ψ , . We may assume η < 14 and then the segment l +3 = (cid:110) z ∈ C (cid:12)(cid:12)(cid:12) arg (cid:16) e √− θ − πm − z − ηse √− π (cid:17) = π m , − η < Re( z ) < (cid:111) intersects with the segment l at a point s e √− πm − θ − πm − + δm − ) satisfying 0 < s < < s if s > 0. Apicture of the region (cid:110) ˜ z = e √− θ − πm − z (cid:12)(cid:12)(cid:12) ( z, s, e √− ψ ) ∈ R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17)(cid:111) looks like [figure 2] NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 33 [figure 1]. .0 . e √− πm + δm − ) . e √− π m . e −√− π m . e √− θ − πm − l . e √− θ − πm − l . s e √− πm + δm − ) . e √− θ − πm − P ( j ) ψ , . angle (2 m +1) δm − [figure 2]. .0 . e √− πm + δm − ) . e √− π m . e −√− π m . e √− θ − πm − l . e √− θ − πm − l . s e √− πm + δm − ) . e √− θ − πm − R ( j ) ψ , . angle (2 m +1) δm − . ηse √− π . e √− θ − πm − l +3 . s e √− πm + δm − ) In the case of (cid:15) = se √− ψ = 0, we can see Q ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) , e √− ψ (cid:1)(cid:9)(cid:17) = ∅ by the definition of Q ( j ) ψ , ,from which we have R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) , e √− ψ (cid:1)(cid:9)(cid:17) = P ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) , e √− ψ (cid:1)(cid:9)(cid:17) . In any case, (cid:0) z , s , e √− ψ (cid:1) lies in R ( j ) ψ , and R ( j ) ψ , ∩ (cid:36) − ( D ) = (cid:110)(cid:0) (cid:15)ζ jm , s, e √− ψ (cid:1) ∈ R ( j ) ψ , (cid:12)(cid:12)(cid:12) (cid:15) = se √− ψ (cid:111) . Consider the differential equation dz ( t ) dt = e √− θ ( z ( t ) m − (cid:15) m ) = e √− θ ( z ( t ) − (cid:15)ζ m )( z ( t ) − (cid:15)ζ m ) · · · ( z ( t ) − (cid:15)ζ mm )with respect to a real time variable t and the initial point z (0) ∈ R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ ). The solution ofthe above differential equation is equivalent to the flow of the vector field v (cid:15),θ = Re (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂x + Im (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂y starting at a point in R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ). Notice that the direction of the vector v (cid:15),θ is given byarg (cid:16) e √− θ ( z ( t ) m − (cid:15) m ) (cid:17) . We investigate the direction of the vector v (cid:15),θ at each boundary point of thefiber R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) of R ( j ) ψ , over ( s, e √− ψ ) ∈ [0 , ) × S .First take a boundary point ( z, s, e √− ψ ) of R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) satisfying arg( z ) + θ − πm − πm + 2 δm − (cid:16) e √− θ − πm − e √− θ z m (cid:17) = θ − πm − θ + m arg( z ) = m ( θ − π ) m − m arg( z ) + π = 2 π + 2 mδm − . Combined with the inequality (35), we have − mδ m − < arg (cid:16) e √− θ − πm − e √− θ ( z m − ( (cid:15)ζ jm ) m ) (cid:17) < mδm − < πm + 2 δm − , from which we can see that the vector v (cid:15),θ faces toward the interior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) .Secondly take a boundary point ( z, s, e √− ψ ) of R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) satisfying arg( z ) + θ − πm − − π m . Then we havearg (cid:16) e √− θ − πm − e √− θ z m (cid:17) = θ − πm − θ + m arg( z ) = m ( θ − π ) m − m arg( z ) + π = 2 π . Combined with (35), we have − π m < − mδ m − < arg (cid:16) e √− θ − πm − e √− θ ( z m − ( (cid:15)ζ jm ) m ) (cid:17) < π . So the vector v (cid:15),θ faces toward the interior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) .Thirdly we take a boundary point ( z, s, e √− ψ ) of R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) which satisfies the equalityarg (cid:16) e √− π m − e √− θ − πm − z (cid:17) = (2 m + 1) δm − z lies on the segment l . Since π π ≤ arg (cid:16) e √− θ − πm − e √− θ z m (cid:17) ≤ π + 2 mδm − − π ≤ arg (cid:16) e √− θ − πm − e √− θ ( z m − ( (cid:15)ζ jm ) m ) (cid:17) ≤ mδm − < (2 m + 1) δm − v (cid:15),θ faces toward the interior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) at this point.A picture of the direction of the vector v (cid:15),θ is [figure 3]. NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 35 Fourthly we take a boundary point (cid:0) z, s, e √− ψ (cid:1) of R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) s, e √− ψ (cid:1)(cid:9)(cid:17) satisfying | z | = 1. Notethat we have − π m ≤ arg( z ) + θ − πm − ≤ π m . If π m ≤ arg( z ) + θ − πm − ≤ π m , then7 π ≤ arg (cid:16) e √− θ − πm − e √− θ z m (cid:17) = θ − πm − θ + m arg( z ) = m ( θ − π ) m − m arg( z ) + π ≤ π . Since | (cid:15) m | ≤ s < 13 = 13 | z | m and 5 π ≤ arg (cid:16) e √− θ − πm − e √− θ (cid:15) m (cid:17) ≤ π π ≤ arg (cid:16) e √− θ − πm − e √− θ ( z m − ( (cid:15)ζ jm ) m ) (cid:17) < π . So the vector v (cid:15),θ faces toward the interior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) s, e √− ψ (cid:1)(cid:9)(cid:17) at this point. If − π m ≤ arg( z ) + θ − πm − ≤ − π m , then we have2 π ≤ arg (cid:16) e √− θ − πm − e √− θ z m (cid:17) = m ( θ − π ) m − m arg( z ) + π ≤ π | (cid:15) m | < = | z m | , a rough estimate π < arg (cid:16) e √− θ − πm − e √− θ ( z m − (cid:15) m ) (cid:17) = m ( θ − π ) m − m arg( z ) + π ≤ π . So the vector v (cid:15),θ faces toward the interior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) s, e √− ψ (cid:1)(cid:9)(cid:17) at this point. If − π m ≤ arg( z ) + θ − πm − ≤ π m , then we have 5 π ≤ arg (cid:16) e √− θ − πm − e √− θ z m (cid:17) ≤ π π < arg (cid:16) e √− θ − πm − e √− θ ( z m − (cid:15) m ) (cid:17) < π | (cid:15) m | < = | z | . So v (cid:15),θ faces toward the interior of the region R ( j ) ψ ∩ (cid:16) ∆ × (cid:8)(cid:0) s, e √− ψ (cid:1)(cid:9)(cid:17) .Finally we take a boundary point (cid:0) z, s, e √− ψ (cid:1) of R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8)(cid:0) s, e √− ψ (cid:1)(cid:9)(cid:17) satisfying (cid:0) z, s, e √− ψ (cid:1) ∈ Q ( j ) ψ , and arg (cid:16) e √− θ − πm − z − ηse √− π (cid:17) = ± π m . Then we have | z | ≤ sη and − π m < − mδm − < arg (cid:16) e √− θ − πm − e √− θ (cid:0) z m − (cid:0) se √− ψ (cid:1) m (cid:1)(cid:17) < mδm − < π m because of the inequality (36) and the assumption 0 < δ < π m . Thus the vector v (cid:15),θ faces toward theinterior of the region R ( j ) ψ , ∩ (cid:16) ∆ × (cid:8) ( s, e √− ψ ) (cid:9)(cid:17) at this point. A picture of the direction of v (cid:15),θ is [figure4]. [figure 3]. .0 . e √− πm + δm − ) . e √− π m . e −√− π m . e √− θ − πm − l . e √− θ − πm − l . s e √− πm + δm − ) . e √− θ − πm − R ( j ) ψ , . angle (2 m +1) δm − . ηse √− π . l +3 . s e √− πm + δm − ) [figure 4]. . e √− πm + δm − ) . e √− π m . e −√− π m . e √− θ − πm − l . e √− θ − πm − l . s e √− πm + δm − ) . e √− θ − πm − R ( j ) ψ , . angle (2 m +1) δm − . ηse √− π . e √− θ − πm − l +3 . s e √− πm + δm − ) NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 37 From all the above arguments, we can see that the flows of the vector field v (cid:15),θ stay inside the region R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ). Take a flow { ( z ( t ) , ( s, e √− ψ )) | t ≥ } inside R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ). If we set R (cid:48) := ( z, s, e √− ψ ) ∈ ∆ × (cid:104) , (cid:17) × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − δ m − < ψ + 2 jπm + θ − πm − < δ m − ,z (cid:54) = 0 , − π m < arg( z ) + θ − πm − < π m , then we have R (cid:48) ⊂ R ( j ) ψ , and we can see by the argument similar to the former analysis on the direction of v (cid:15),θ that flows of v (cid:15),θ starting at points in R (cid:48) \ ( (cid:36) − ( D ) ∩ R (cid:48) ) stay inside R (cid:48) \ ( (cid:36) − ( D ) ∩ R (cid:48) ). Take any point( z, s, e √− ψ ) ∈ R ( j ) ψ , \ R (cid:48) . If z (cid:54) = 0, then we have either ( z, s, e √− ψ ) ∈ Q ( j ) ψ , or π m < arg( z ) + θ − πm − <πm + 2 δm − | z | < ηs or(37) 4 π < arg (cid:16) e √− θ − πm − e √− θ z m (cid:17) < π + 2 mδm − . Combined with (35), we have e √− θ ( z m − (cid:15) m ) (cid:54) = 0. If z = 0, then s > e √− θ ( z m − (cid:15) m ) (cid:54) = 0again. So v (cid:15),θ does not vanish on R ( j ) ψ , \ R (cid:48) and there is no limit point lim t →∞ z ( t ) inside R ( j ) ψ , \ R (cid:48) . Sincethe inequality (37) holds as long as ( z, s, e √− ψ ) lies in P ( j ) ψ , \ R (cid:48) , flows of v (cid:15),θ do not stay inside R ( j ) ψ , \ R (cid:48) and there exists t > z ( t ) , s, e √− ψ ) is contained in the region R (cid:48) \ ( (cid:36) − ( D ) ∩ R (cid:48) ).If ( z ( t ) , ( s, e √− ψ )) ∈ R (cid:48) \ ( (cid:36) − ( D ) ∩ R (cid:48) ), then we have − ( m − π m ≤ arg (cid:32) m − (cid:88) l =0 (cid:16) z ( t ) e √− θ − πm − (cid:17) m − − l (cid:16) e √− θ − πm − (cid:15)ζ jm (cid:17) l (cid:33) ≤ ( m − π m . By the calculation ddt | z ( t ) − (cid:15)ζ jm | m = 1( z ( t ) − (cid:15)ζ jm ) m ddt (cid:32) z ( t ) − (cid:15)ζ jm ) m (cid:33) + 1( z ( t ) − (cid:15)ζ jm ) m ddt (cid:32) z ( t ) − (cid:15)ζ jm ) m (cid:33) = 1( z ( t ) − (cid:15)ζ jm ) m − m ( z ( t ) − (cid:15)ζ jm ) m +1 dz ( t ) dt + 1( z ( t ) − (cid:15)ζ jm ) m − m ( z − (cid:15)ζ jm ) m +1 dz ( t ) dt = − me √− θ ( z ( t ) m − ( (cid:15)ζ jm ) m )( z ( t ) − (cid:15)ζ jm ) m +1 ( z ( t ) − (cid:15)ζ jm ) m − me √− θ ( z ( t ) m − ( (cid:15)ζ jm ) m )( z ( t ) − (cid:15)ζ jm ) m ( z ( t ) − (cid:15)ζ jm ) m +1 = 2 m | z ( t ) − (cid:15)ζ jm | m Re (cid:32) − e √− θ z ( t ) m − ( (cid:15)ζ jm ) m z ( t ) − (cid:15)ζ jm (cid:33) = 2 m Re (cid:16) − e √− θ (cid:0) z ( t ) m − + (cid:15)ζ jm z ( t ) m − + · · · + ( (cid:15)ζ jm ) m − z ( t ) + ( (cid:15)ζ jm ) m − (cid:1)(cid:17) | z ( t ) − (cid:15)ζ jm | m , we can see ddt | z ( t ) − (cid:15)ζ jm | m = 2 m Re (cid:16) e √− θ − π ) (cid:0) z ( t ) m − + (cid:15)ζ jm z ( t ) m − + · · · + ( (cid:15)ζ jm ) m − z ( t ) + ( (cid:15)ζ jm ) m − (cid:1)(cid:17) | z ( t ) − (cid:15)ζ jm | m = 2 m | z ( t ) − (cid:15)ζ jm | m Re (cid:32) m − (cid:88) l =0 (cid:16) z ( t ) e √− θ − πm − (cid:17) m − − l (cid:16) e √− θ − πm − (cid:15)ζ jm (cid:17) l (cid:33) ≥ m | z ( t ) − (cid:15)ζ jm | m (max {| z ( t ) | , | (cid:15) |} ) m − cos (cid:18) ( m − π m (cid:19) ≥ m | z ( t ) − (cid:15)ζ jm | m (cid:18) | z ( t ) − (cid:15)ζ jm | (cid:19) m − m m − | z ( t ) − (cid:15)ζ jm | m +1 ≥ m m > . So we have 1 | z ( t ) − (cid:15)ζ jm | m ≥ m m t − C for some constant C > 0. Thus we havelim t →∞ z ( t ) = (cid:15)ζ jm . and the flow of v θ,(cid:15) starting at any point of R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ) converges to ( (cid:15)ζ jm , s, e √− ψ ) ∈ (cid:36) − ( D ). Case 2. − πm ≤ arg( z ) − jπm − ψ < δ > δ < π m and put(38) θ ( i ) ψ , := − j ( m − πm − ( m − ψ + π − δ. If we simply write θ := θ ( i ) ψ , , then we have − πm − δm − ≤ arg( z ) + θ − πm − ≤ − δm − . We take < s < η > P ( j ) ψ , = ( z, ( s, e √− ψ )) ∈ ∆ × (cid:104) , (cid:17) × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − δ m − < ψ + 2 jπm + θ − πm − < δ m − , z (cid:54) = 0 − π − π m < arg (cid:16) e −√− π m − e √− θ − πm − z (cid:17) < − (2 m + 1) δm − − πm − δm − < arg( z ) + θ − πm − < π m Q ( j ) ψ , := ( z, s, e √− ψ ) ∈ ∆ × (cid:104) , (cid:17) × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e √− θ − πm − z − ηse √− π (cid:54) = 0 , − π m < arg (cid:16) e √− θ − πm − z − ηse √− π (cid:17) < π m , − δ m − < ψ + 2 kπm + θ − πm − < δ m − π m ≤ arg( z ) + θ − πm − ≤ π − πm − δm − z (cid:54) = 0 .R ( j ) ψ , := P ( j ) ψ , ∪ Q ( j ) ψ , . By the similar argument to Case 1, we can see that ( z , s , e √− ψ ) ∈ R ( j ) ψ , and the flow ( z ( t ) , s, e √− ψ ) t ≥ of v (cid:15),θ starting at a point in R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ) satisfieslim t →∞ z ( t ) = (cid:15)ζ jm . If we put U := ( { } × { } × S ) ∪ (cid:91) R ( j ) ψ , , then we can see by the construction of R ( j ) ψ , that (cid:8) z ∈ ∆ (cid:12)(cid:12) | z | < (cid:9) × (cid:2) , (cid:1) × S is contained in U . So wecan write U = (cid:18)(cid:110) z ∈ ∆ (cid:12)(cid:12)(cid:12) | z | < (cid:111) × (cid:104) , (cid:17) × S (cid:19) ∪ (cid:91) R ( j ) ψ , and we can see that U is an open neighborhood of { } × { } × S in ∆ × [0 , × S . If we put W ( j ) ψ , := R ( j ) ψ , \ ( (cid:36) − ( D ) ∩ R ( j ) ψ , ) , then we have an open covering U \ ( U ∩ (cid:36) − ( D )) = (cid:91) W ( j ) ψ , . This covering satisfies the statement of the proposition. (cid:3) NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 39 Fundamental solution with an asymptotic property. We use the same notations as in subsection3.1 Take a point p ∈ W ( j ) ψ ,ξ and consider the holomorphic solution (cid:0) z ( τ ) , s, e √− ψ (cid:1) of the differentialequation(39) dz ( τ ) dτ = e √− θ ( z ( τ ) m − (cid:15) m )satisfying (cid:0) z (0) , s, e √− ψ (cid:1) = p , where (cid:15) = se √− ψ and θ = θ ( j ) ψ ,ξ . If we take t , u ∈ R and if we fix t + √− u constant, (cid:0) z ( t + t + √− u ) , s, e √− ψ (cid:1) t ≥ coincides with the flow (cid:0) z t + √− u ( t ) , s, e √− ψ (cid:1) of v (cid:15),θ satisfying z t + √− u (0) = z ( t + √− u ). So we can extend the solution ( z ( τ ) , s, e √− ψ ) by an analyticcontinuation to a holomorphic function in τ on an open neighborhood of R ≥ whose image by z ( τ ) is anopen neighborhood of the flow of v (cid:15),θ starting at the point p . Note that we havelim t →∞ z ( t + √− u ) = (cid:15)ζ jm and z t + √− u ( t ) = z ( t + t + √− u ) = z √− u ( t + t ).The following theorem is a weak unfolded analogue of the existence theorem of fundamental solutionswith an asymptotic property [43, Theorem 12.1] in the irregular singular case. It is an easy restrictedcase of a more general theorem in [14] and [15], which is one of the main tools in the unfolding theory byHurtubise, Lambert and Rousseau. Theorem 3.2 ([14, Theorem 5.3], [15, Theorem 2.5]) . Consider the linear differential equation (40) df dz...df r dz = A ( z, (cid:15), w )( z m − (cid:15) m ) f ...f r on the polydisk ∆ × ∆ × ∆ s , where A ( z, (cid:15), w ) is an r × r matrix of holomorphic functions in ( z, (cid:15), w ) =( z, (cid:15), w , . . . , w s ) ∈ ∆ × ∆ × ∆ s such that A ( z, (cid:15), w ) − ν ( z, (cid:15), w ) · · · ... . . . ... · · · ν r ( z, (cid:15), w ) ∈ ( z m − (cid:15) m ) M r ( O hol ∆ × ∆ × ∆ s ) , where ν ( z, (cid:15), w ) , . . . , ν r ( z, (cid:15), w ) are polynomials in z whose coefficients are holomorphic functions in (cid:15), w and ν ( (cid:15)ζ jm , (cid:15), w ) , . . . , ν r ( (cid:15)ζ jm , (cid:15), w ) are mutually distinct for any fixed j , (cid:15) and w . Then for a certain choiceof the open covering { W ( j ) ψ ,ξ } of U \ ( (cid:36) − ( D ) ∩ U ) in Proposition 3.1, there are an open covering W ( j ) ψ ,ξ × ∆ s = (cid:91) p ∈ W ( j ) ψ ,ξ S ( j ) ψ ,ξ,p , and a matrix Y ϑ ( z ) = (cid:0) y ϑ ( z ) , . . . , y ϑr ( z ) (cid:1) of solutions on S ϑ := S ( j ) ψ ,ξ,p of the differential equation (40), thatis, d Y ϑ ( z ) dz = A ( z, (cid:15), w ) z m − (cid:15) m Y ϑ ( z ) such that for the solution z ( τ ) of the holomorphic differential equation (39) with the initial value z (0) = p ∈ S ( j ) ψ ,ξ,p , the limit lim t →∞ Y ϑ ( z ( t + u )) exp − (cid:82) tt ν ( z ( t + u )) e √− θ dt · · · ... . . . ... · · · (cid:82) tt ν r ( z ( t + u )) e √− θ dt = C ϑu ( s, e √− ψ , w ) along the flow ( z ( t + u )) t ≥ exists and the limit C ϑu ( se √− ψ , w ) is a diagonal matrix of functions continuousin s, e √− ψ , w, t , u and holomorphic in w and (cid:15) = se √− ψ (cid:54) = 0 . Proof. For the solution z ( τ ) of the differential equation (39) with an initial value ( z (0) , s, e √− ψ ) = p in W ( j ) ψ ,ξ , we consider z ( t + u ) for u ∈ C with | u | (cid:28) 1. If we write (cid:15) := se √− ψ , the restriction of the differentialequation (40) to the flow z ( t + u ) of v (cid:15),θ becomes df ( z ( t + u ) , (cid:15), w ) dt...df r ( z ( t + u ) , (cid:15), w ) dt = e √− θ A ( z ( t + u ) , (cid:15), w ) f ( z ( t + u ) , (cid:15), w ) ...f r ( z ( t + u ) , (cid:15), w ) . Since the flow ( z ( t + u ) , s, e √− ψ , w ) is contained in W ( j ) ψ ,ξ × ∆ s , we have lim t →∞ z ( t + u ) = (cid:15)ζ jm andlim t →∞ e √− θ A ( z ( t + u ) , (cid:15), w ) = e √− θ A ( (cid:15)ζ jm , (cid:15), w )= e √− θ ν ( (cid:15)ζ jm , (cid:15), w ) · · · ... . . . ... · · · e √− θ ν r ( (cid:15)ζ jm , (cid:15), w ) . We may assume by a suitable choice of δ > θ = θ ( j ) ψ ,ξ in (34) and (38) that the real partsRe (cid:16) e √− θ ν ( (cid:15)ζ jm , (cid:15), w ) (cid:17) , . . . , Re (cid:16) e √− θ ν r ( (cid:15)ζ jm , (cid:15), w ) (cid:17) of the eigenvalues of the matrix e √− θ A ( (cid:15)ζ jm , (cid:15), w )are mutually distinct. Moreover we may assume by replacing the order of a holomorphic frame that(41) Re (cid:16) e √− θ ν ( (cid:15)ζ jm , (cid:15), w ) (cid:17) < · · · < Re (cid:16) e √− θ ν r ( (cid:15)ζ jm , (cid:15), w ) (cid:17) holds. As in the proof of Proposition 3.1, we have − ( m − π m ≤ arg (cid:18)(cid:16) e √− θ − πm − z ( t + u ) (cid:17) m − (cid:19) ≤ ( m − π m for sufficiently large t > 0. So we have ddt | z ( t + u ) m − (cid:15) m | = 12( | z ( t + u ) m − (cid:15) m | ) ddt (cid:16) ( z ( t + u ) m − (cid:15) m )( z ( t + u ) m − (cid:15) m ) (cid:17) = 2 Re (cid:16) mz ( t + u ) m − z (cid:48) ( t + u )( z ( t + u ) m − (cid:15) m ) (cid:17) | z ( t + u ) m − (cid:15) m | = Re (cid:16) me √− θ z ( t + u ) m − ( z ( t + u ) m − (cid:15) m )( z ( t + u ) m − (cid:15) m ) (cid:17) | z ( t + u ) m − (cid:15) m | = Re (cid:18) − m (cid:16) e √− θ − πm − z ( t + u ) (cid:17) m − (cid:19) | z ( t + u ) m − (cid:15) m |≤ − m cos (cid:18) ( m − π π (cid:19) (cid:12)(cid:12) z ( t + u ) m − (cid:12)(cid:12) | z ( t + u ) m − (cid:15) m |≤ − m | z ( t + u ) m − (cid:15) m | m − m | z ( t + u ) m − (cid:15) m | for sufficiently large t > 0, from which we have ddt (cid:16) | z ( t + u ) m − (cid:15) m | − m − m (cid:17) = − m − m | z ( t + u ) m − (cid:15) m | − m − m − ddt | z ( t + u ) m − (cid:15) m |≥ m − . So there exists a constant C > | z ( t + u ) m − (cid:15) m | − m − m ≥ m − t − C holds for sufficiently large t > 0. If we write ν k = (cid:80) ql =0 b l ( (cid:15), w ) z l , we have ddt e √− θ ν k ( z ( t + u ) , (cid:15), w ) = e √− θ q (cid:88) l =0 l b l ( (cid:15), w ) z ( t + u ) l − e √− θ ( z ( t + u ) m − (cid:15) m ) . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 41 So there is a constant C (cid:48) > (cid:12)(cid:12)(cid:12)(cid:12) ddt e √− θ ν j ( z ( t + u ) , (cid:15), w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:48) | z ( t + u ) m − (cid:15) m | and (cid:90) ∞ a (cid:12)(cid:12)(cid:12)(cid:12) ddt e √− θ ν k ( z ( t + u ) , (cid:15), w ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ C (cid:48) (cid:90) ∞ a | z ( t + u ) m − (cid:15) m | dt ≤ C (cid:48) (cid:90) ∞ a (cid:18) m − t − C (cid:19) − − m − dt < ∞ for a reference point a ∈ R > . Similarly we have (cid:90) ∞ a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ( z ( t + u ) , (cid:15), w ) − ν ( z ( t + u ) , (cid:15), w ) · · · ... . . . ... · · · ν r ( z ( t + u ) , (cid:15), w ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) dt < ∞ because the absolute values of the entries of the matrix A ( z ( t + u ) , (cid:15), w ) − ν ( z ( t + u ) , (cid:15), w ) · · · ... . . . ... · · · ν r ( z ( t + u ) , (cid:15), w ) are bounded by C (cid:48)(cid:48) | z ( t + u ) m − (cid:15) m | for some constant C (cid:48)(cid:48) > 0. Thus, by the theorem of Levinson ([31,Theorem 1]), there are t > Y u ( t, s, e √− ψ , w ) = (cid:16) y u ( t, s, e √− ψ , w ) , . . . , y ur ( t, s, e √− ψ , w ) (cid:17) of solutions y u ( t, s, e √− ψ , w ) , . . . , y ur ( t, s, e √− ψ , w ) of the differential equation(42) dy ( t ) dt = e √− θ A ( z ( t + u ) , (cid:15), w ) y ( t )defined for t > t − b for some b > 0, which satisfieslim t →∞ Y u ( t, s, e √− ψ , w ) exp − (cid:82) tt ν ( z ( t + u )) e √− θ dt · · · ... . . . ... · · · (cid:82) tt ν r ( z ( t + u )) e √− θ dt (43) = C u ( s, e √− ψ , w ) = c ( u ) · · · ... . . . ... · · · c r ( u ) with C u ( (cid:15), w ) constant in z satisfying A ( (cid:15)ζ jm , (cid:15), w ) C u ( s, e √− ψ , w ) = C u ( s, e √− ψ , w ) ν ( (cid:15)ζ jm , (cid:15), w ) · · · ... . . . ... · · · ν r ( (cid:15)ζ jm , (cid:15), w ) . Notice that y uk ( t, s, e √− ψ , w ) is constructed in [31] by applying an infinite sum and integrations of theform (cid:82) ta or (cid:82) ∞ t to given functions in t, s, e √− ψ , w, u constructed from A ( z, (cid:15), w ). So we can see by theirconstruction in [31] that the solutions y uk ( t, s, e √− ψ , w ) are functions continuous in s, e √− ψ , w, u andholomorphic in w, u and (cid:15) (cid:54) = 0. Furthermore, C u ( s, e √− ψ , w ) is a matrix of functions continuous in s, e √− ψ , w, u and holomorphic in w , u and (cid:15) (cid:54) = 0. Since A ( (cid:15)ζ jm , (cid:15), w ) is a diagonal matrix with the distincteigenvalues by the assumption, C u ( (cid:15), w ) becomes a diagonal matrix.By the fundamental theorem of ordinary linear differential equations, there exists a fundamental solution Y ϑ ( z, s, e √− ψ , w ) = (cid:16) y ϑ ( z, s, e √− ψ , w ) , . . . , y ϑr ( z, s, e √− ψ , w ) (cid:17) of the differential equation (40), that is to say, dY ϑ dz = A ( z ) z m − s m e √− mψ Y ϑ in a neighborhood of ( z ( t ) , s, e √− ψ , w ) which satisfies the initial condition Y ϑ ( z ( t ) , s, e √− ψ , w ) = Y ( t , s, e √− ψ , w ) . Here the suffix ϑ means the data p, j, ψ , ξ . Since the solutions of the linear differential equation (40)form a local system on U \ ( D ∩ U ), we can extend Y ϑ ( z ) to a matrix of holomorphic functions in aneighborhood of { z ( t ) | t ≥ t } by an analytic continuation. We fix u ∈ C close to the origin 0. Since both Y ϑ ( z ( t + u ) , s, e √− ψ , w ) := (cid:16) y ϑ ( z ( t + u ) , s, e √− ψ , w ) , . . . , y ϑr ( z ( t + u ) , s, e √− ψ , w ) and Y u ( t, s, e √− ψ , w )satisfy the same linear differential equation dYdt = e √− θ A ( z ( t + u )) Y, there is a matrix P ( u ) of functions continuous in s, e √− ψ , w, u and holomorphic in w and u satisfying Y ϑ ( z ( t + u ) , s, e √− ψ , w ) = Y u ( t, s, e √− ψ , w ) P ( u )for t close to t . We put Λ k ( t, u ) := exp (cid:16)(cid:82) tt ν k ( z ( t + u )) e √− θ dt (cid:17) . By (41), lim t →∞ Λ k ( t ) − Λ k (cid:48) ( t ) isdivergent if k < k (cid:48) . If u ∈ R is a real number, we can see by the property (43) for u = 0 thatlim t →∞ Y ϑ ( z ( t + u )) exp − (cid:82) tt ν ( z ( t + u )) e √− θ dt · · · ... . . . ... · · · − (cid:82) tt ν r ( z ( t + u )) e √− θ dt = lim t →∞ Y ϑ ( z ( t + u )) exp − (cid:82) t + ut + u ν ( z ( t (cid:48) )) e √− θ dt (cid:48) · · · ... . . . ... · · · − (cid:82) t + ut + u ν r ( z ( t (cid:48) )) e √− θ dt (cid:48) = C u ( s, e √− ψ , w ) exp (cid:16)(cid:82) t + ut ν ( z ( t (cid:48) )) e √− θ dt (cid:48) (cid:17) · · · ... . . . ... · · · exp (cid:16)(cid:82) t + ut ν r ( z ( t (cid:48) )) e √− θ dt (cid:48) (cid:17) is convergent and its limit is a diagonal matrix. If we put Y u ( t ) = (cid:0) y u ( t ) , . . . , y ur ( t ) (cid:1) , P ( u ) = p , ( u ) · · · p ,r ( u ) ... . . . ...p r, ( u ) · · · p r,r ( u ) , then, for u ∈ R , Y ϑ ( z ( t + u )) Λ ( t ) − · · · ... . . . ... · · · Λ r ( t ) − = Y u ( t ) P ( u ) Λ ( t ) − · · · ... . . . ... · · · Λ r ( t ) − = (cid:0) y u ( t ) , . . . , y ur ( t ) (cid:1) p , ( u ) · · · p ,r ( u ) ... . . . ...p r, ( u ) · · · p r,r ( u ) Λ ( t ) − · · · ... . . . ... · · · Λ r ( t ) − = (cid:32) r (cid:88) k =1 p k, ( u )Λ ( t ) − y uk ( t ) , . . . , r (cid:88) k =1 p k,r ( u )Λ r ( t ) − y uk ( t ) (cid:33) is bounded when t → ∞ . Note thatΛ l ( t ) − y uk ( t ) = (cid:0) Λ l ( t ) − Λ k ( t ) (cid:1) (cid:0) Λ k ( t ) − y uk ( t ) (cid:1) is divergent for l < k when t → ∞ , because lim t →∞ Λ l ( t ) − Λ k ( t ) is divergent and lim t →∞ Λ k ( t ) − y uk ( t ) = c k ( u ) e k (cid:54) = 0. So we should have p k,l ( u ) = 0 for k > l and u ∈ R with | u | (cid:28) 1. Since p k,l ( u ) is holomorphicin u , we have p k,l ( u ) = 0 for u ∈ C with | u | (cid:28) 1. In other words, P ( u ) is an upper triangular matrix of NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 43 holomorphic functions in u . Then we have, for u ∈ C with | u | (cid:28) 1, thatlim t →∞ Y ϑ ( z ( t + u )) exp − (cid:82) tt ν ( z ( s + u )) e √− θ ds · · · ... . . . ... · · · − (cid:82) tt ν r ( z ( s + u )) e √− θ ds = lim t →∞ (cid:0) y u ( t ) , . . . , y ur ( t ) (cid:1) p , ( u ) · · · p ,r ( u ) ... . . . ... · · · p r,r ( u ) Λ ( t ) − · · · ... . . . ... · · · Λ r ( t ) − converges to a diagonal matrix C ϑu ( s, e √− ψ , w ). (cid:3) Remark 3.3. Although a formal solution transforming an unfolded linear differential equation to a normalform is given in [14, Theorem 3.2], we cannot expect to construct a fundamental solution of (40) with anasymptotic property with respect to the formal solution as in the irregular singular case ([43, Theorem12.1]). 4. Construction of a local horizontal lift In this section, we construct an integrable connection which is a first order infinitesimal extension of agiven local relative connection. We call this extension a local horizontal lift, or a block of local horizontallifts in section 5, which is a key part in the construction of an unfolding of the unramified irregular singulargeneralized isomonodromic deformation. A basic idea in this section is to extend a local connection to aglobal connection on P with regular singularity at ∞ . Unfortunately, our construction of a local horizontallift is not canonical but it is systematically determined. So it enables us to construct a non-canonical globalhorizontal lift in section 5, which induces an unfolded generalized isomonodromic deformation.4.1. Extension of a local connection to a global connection on P . Consider the divisor D := { ( z, (cid:15), w ) ∈ ∆ × ∆ × ∆ s | z m − (cid:15) m = 0 } on the polydisk ∆ × ∆ × ∆ s , where ∆ = { z ∈ C | | z | < } . If we put D j := (cid:8) ( z, (cid:15), w ) ∈ ∆ × ∆ × ∆ s (cid:12)(cid:12) z − (cid:15)ζ jm = 0 (cid:9) for j = 1 , . . . , m with ζ m = exp( π √− m ), then we can write D = D + · · · + D m as an effective divisor on ∆ × ∆ × ∆ s . We consider a family of intervalsΓ ∆ ,j = (cid:8) ( sζ jm (cid:15), (cid:15), w ) ∈ ∆ × ∆ × ∆ s (cid:12)(cid:12) ≤ s ≤ (cid:9) which join the origin 0 and ζ jm (cid:15) and consider their unionΓ ∆ := m (cid:91) j =1 Γ ∆ ,j . We consider the embedding ∆ × ∆ × ∆ s (cid:44) → P × ∆ × ∆ s = P × ∆ s and regard D as an effective divisoron P × ∆ × ∆ s .We prepare a notation of diagonal matrix. Notation 4.1. We denote the diagonal matrix whose ( k, k ) entry is a k by Diag ( a k ) ;Diag ( a k ) = a · · · ... . . . ... · · · a r . Take mutually distinct complex numbers µ , . . . , µ r and a polynomial ν ( T ) ∈ O D [ T ] given by ν ( T ) = r − (cid:88) l =0 (cid:16) m − (cid:88) j =0 c l,j z j (cid:17) T l (44)with c l,j ∈ O ∆ × ∆ s such that ν ( µ ) | p , . . . , ν ( µ r ) | p are distinct complex numbers at any point p ∈ D . We denote the closed interval { t ∈ R | ≤ t ≤ } by [0 , γ : [0 , × ∆ × ∆ s −→ ∆ × ∆ × ∆ s and an open subset W ⊂ ∆ × ∆ × ∆ s such that ˜ γ (0 , b ) = ˜ γ (1 , b ) for any b ∈ ∆ × ∆ s , each fiber W b over b ∈ ∆ × ∆ s is a disk containing D b and that the boundary ∂W b coincides with the image ˜ γ ([0 , × { b } ).Let(45) ∇ ∆ : O ⊕ r ∆ × ∆ × ∆ s (cid:51) f ...f r (cid:55)→ df ...df r + A ( z, (cid:15), w ) dzz m − (cid:15) m f ...f r ∈ Ω × ∆ × ∆ s / ∆ × ∆ s ( D ) ⊕ r be a relative connection on ∆ × ∆ × ∆ s over ∆ × ∆ s satisfying(46) A ( z, (cid:15), w ) (cid:12)(cid:12) D = Diag ( ν ( µ k )) (cid:12)(cid:12) D = ν ( µ ) · · · ... . . . ... · · · ν ( µ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D . For each point b ∈ ∆ × ∆ s , we consider the restriction ∇ ∆ b := ∇ ∆ | ∆ ×{ b } and its associated connection ∇ † ∆ b : E nd (cid:0) O ⊕ r ∆ ×{ b } (cid:1) (cid:51) u (cid:55)→ ∇ ∆ b ◦ u − u ◦ ∇ ∆ b ∈ E nd (cid:0) O ⊕ r ∆ ×{ b } (cid:1) ⊗ Ω ×{ b } ( D b ) . We assume the following condition for ∇ ∆ : Assumption 4.2. (i) the monodromy of ∇ ∆ b along ˜ γ b has a diagonal representation matrix of holo-morphic functions over ∆ × ∆ s with r distinct eigenvalues for any b ∈ ∆ × ∆ s and (ii) H (cid:0) ∆ × { b } , ker (cid:0) ∇ † ∆ b (cid:1)(cid:1) = C for each b ∈ ∆ × ∆ s . Proposition 4.3. There exist an open neighborhood V of (0 , in ∆ × ∆ s and a relative connection ∇ P : (cid:0) O hol P ×V (cid:1) ⊕ r −→ (cid:0) O hol P ×V (cid:1) ⊕ r ⊗ Ω P ×V / V (cid:0) ( D ∩ (∆ × V )) ∪ ( {∞} × V ) (cid:1) hol on P × V over V admitting poles along ( D ∩ (∆ × V )) ∪ ( {∞} × V ) such that the restriction ∇ P | ∆ ×V isisomorphic to the restriction ∇ ∆ | ∆ ×V of ∇ ∆ in (45).Proof. Let Mon ˜ γ ( ∇ ∆ ) be the monodromy matrix of ∇ ∆ along ˜ γ with respect to a local basis of ker ∇ ∆ .We can take a contractible open subset W (cid:48) ⊂ ∆ × ∆ × ∆ s with W (cid:48) ⊂ W such that the fiber W (cid:48) b is a closeddisk for each b ∈ ∆ × ∆ s and that the fundamental group π ((∆ × ∆ × ∆ s ) \ W (cid:48) , ∗ ) is isomorphic to Z which is generated by ˜ γ . We can take a regular singular relative connection ∇ ∞ : (cid:0) O hol P × ∆ × ∆ s \ W (cid:48) (cid:1) ⊕ r −→ (cid:0) O hol P × ∆ × ∆ s \ W (cid:48) (cid:1) ⊕ r ⊗ Ω P × ∆ × ∆ s \ W (cid:48) ) / ∆ × ∆ s ( {∞} × ∆ × ∆ s )such that the monodromy of ∇ ∞ along ˜ γ is given by Mon ˜ γ ( ∇ ∆ ) and that the set of eigenvalues of res ( ∞ ,b (cid:48) ) (cid:0) ∇ ∞ (cid:12)(cid:12) ( P ×{ b (cid:48) } ) \ ( W ∩ ( P ×{ b (cid:48) } )) (cid:1) is contained in { z ∈ C | ≤ Re( z ) < } for any b (cid:48) ∈ ∆ × ∆ s . Notethat (cid:0)(cid:0) O hol (∆ × ∆ × ∆ s ) \ W (cid:48) (cid:1) ⊕ r , ∇ ∆ (cid:12)(cid:12) (∆ × ∆ × ∆ s ) \ W (cid:48) (cid:1) and (cid:0)(cid:0) O hol (∆ × ∆ × ∆ s ) \ W (cid:48) (cid:1) ⊕ r , ∇ ∞ (cid:12)(cid:12) (∆ × ∆ × ∆ s ) \ W (cid:48) (cid:1) are isomorphic,because their corresponding representations of the fundamental group π (cid:0) (∆ × ∆ × ∆ s ) \ W (cid:48) , ∗ (cid:1) ∼ = Z aregiven by the same monodromy matrix Mon ˜ γ ( ∇ ∆ ). So we can patch ∇ ∞ , ∇ ∆ | ∆ × ∆ × ∆ s and obtain a globalrelative connection ∇ : E −→ E ⊗ Ω P × ∆ × ∆ s ) / ∆ × ∆ s (cid:0) D ∪ (cid:0) {∞} × ∆ × ∆ s (cid:1)(cid:1) on P × ∆ × ∆ s over ∆ × ∆ s . We can write E | P ×{ (0 , } ∼ = r (cid:77) k =1 O P ( a k )with a ≥ a ≥ · · · ≥ a r . Assume that a > a r . For some choice of k , the projection ψ (cid:48) : E −→ E | {∞}× ∆ × ∆ s = r (cid:77) k =1 ker (cid:16) ∇ | {∞}× ∆ × ∆ s − ν ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) {∞}× ∆ × ∆ s (cid:17) −→ ker (cid:16) ∇ | {∞}× ∆ × ∆ s − ν ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) {∞}× ∆ × ∆ s (cid:17) NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 45 satisfies ψ (cid:48) | { ( ∞ , (0 , } ( O P ( a )) = ker (cid:16) ∇ | { ( ∞ , (0 , } − ν ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) { ( ∞ , (0 , } (cid:17) ∼ = O { ( ∞ , (0 , } . Thenthere is an open neighborhood V of (0 , 0) in ∆ × ∆ s such that ψ := ψ (cid:48) | P ×V : E | P ×V −→ ker (cid:16) ∇ | {∞}×V − ν ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) {∞}×V (cid:17) is surjective. If we put ( E , ∇ ) := (ker ψ , ∇ | ker ψ ), then ∇ is a relative connection on P × V over V admitting poles along ( D ∩ (∆ × V )) ∪ ( {∞} × V ) and we have E | P ×{ (0 , } ∼ = O P ( a − ⊕ r (cid:77) k =2 O P ( a k ) . Similarly we can choose a surjection ψ : E −→ O {∞}×V after shrinking V such that ker ψ is preserved by ∇ and that ψ ( O (˜ a )) = O {∞}×V for ˜ a := max { a − , a } . Then we put ( E , ∇ ) := (ker ψ , ∇ | ker ψ ).Repeating this procedure, we finally obtain ( E N , ∇ N ) such that E N | P ×V ∼ = O P V ( N ) ⊕ r . So the connection ∇ N ⊗ O ( − N ) satisfies the condition of the proposition. (cid:3) The construction of a local horizontal lift. We use the same notations as in subsection 4.1.We consider the non-reduced analytic space P × ∆ × ∆ s × Spec C [ h ] / ( h ). For an analytic open subset U ⊂ P × ∆ × ∆ s , we denote by U [¯ h ] the analytic open subspace of P × ∆ × ∆ s × Spec C [ h ] / ( h ) whoseunderlying set of points coincides with U . In this subsection, we will construct an extension of the relativeconnection ∇ P constructed in Proposition 4.3 to an integrable connection on P × V [¯ h ] over V . Thisproduces a block of local horizontal lifts defined in Definition 5.8, which is a key concept in the constructionof a global horizontal lift in subsection 5.3.Recall that the sheaf of holomorphic differential forms (cid:0) Ω ( P × ∆ s \ Γ ∆ ) [¯ h ] / ∆ × ∆ s (cid:1) hol on (cid:0) P × ∆ s \ Γ ∆ (cid:1) [¯ h ] isgiven by (cid:0) Ω ( P × ∆ s \ Γ ∆ ) [¯ h ] / ∆ × ∆ s (cid:1) hol = I hol ∆ ( P × ∆ s \ Γ∆ ) [¯ h ] / ∆ × ∆ s (cid:14)(cid:0) I hol ∆ ( P × ∆ s \ Γ∆ ) [¯ h ] / ∆ × ∆ s (cid:1) , where I hol ∆ ( P × ∆ s \ Γ∆ ) [¯ h ] / ∆ × ∆ s is the ideal sheaf of O hol ( P × ∆ s \ Γ ∆ ) [¯ h ] × ∆ × ∆ s ( P × ∆ s \ Γ ∆ ) [¯ h ] which defines the diag-onal (cid:0) P × ∆ s \ Γ ∆ (cid:1) [¯ h ] (cid:44) → (cid:0) P × ∆ s \ Γ ∆ (cid:1) [¯ h ] × ∆ × ∆ s (cid:0) P × ∆ s \ Γ ∆ (cid:1) [¯ h ] . Let ι ( P × ∆ s \ Γ ∆ ) [¯ h ] : (cid:0) P × ∆ s \ Γ ∆ (cid:1) [¯ h ] (cid:44) → P × ∆ s [¯ h ]be the inclusion. We put V [¯ h ] := V × Spec C [ h ] / ( h ). We denote D × ∆ × ∆ s V , Γ × ∆ × ∆ s V by D V , Γ V ,respectively and denote D × ∆ × ∆ s V [¯ h ] by D V [¯ h ]. We first construct an extension of the relative connection ∇ P to a relative connection on P × V [¯ h ] over V [¯ h ]. We need the following lemma: Lemma 4.4. Let A , . . . , A m be elements of End C ( C r ) satisfying m (cid:92) j =1 ker ad( A j ) = C · id , where ad( A j ) : End C ( C r ) (cid:51) X (cid:55)→ A j X − XA j ∈ End C ( C r ) is the adjoint map. Then we have m (cid:88) j =1 im(ad( A j )) = ker (cid:16) End C ( C r ) Tr −→ C (cid:17) . Proof. In general we have t ad( A j ) = − ad( A j ), becauseTr( t ad( A j )( X ) · B ) = Tr( X · ad( A j )( B )) = Tr( X · ( A j B − BA j ))= Tr(( XA j − A j X ) B + A j XB − XBA j )= Tr(( XA j − A j X ) B ) + Tr( A j XB ) − Tr( XBA j )= Tr( − ad( A j )( X ) · B )for any X, B ∈ End C ( C r ). So there are exact sequences0 −→ ker ad( A j ) −→ End C ( C r ) ad( A j ) −−−−→ End C ( C r ) −→ (ker ad( A j )) ∨ −→ . for j = 1 , . . . , m . Since End C ( C r ) π −→ End C ( C r ) (cid:14) (cid:80) mj =1 im(ad( A j )) is the largest quotient vector spacesatisfying π ◦ ad( A j ) = 0 for j = 1 , . . . , m , its dual is given by (cid:16) End C ( C r ) (cid:46) m (cid:88) j =1 im(ad( A j )) (cid:17) ∨ = m (cid:92) j =1 ker t ad( A j ) = m (cid:92) j =1 ker ad( A j ) = C · id ⊂ End C ( C r ) . Taking the dual again, we obtainEnd C ( C r ) (cid:46) m (cid:88) j =1 im(ad( A j )) = ( C · id) ∨ = End C ( C r ) (cid:14) ker (cid:0) End C ( C r ) Tr −→ C ) (cid:1) . Thus we have (cid:80) mj =1 im(ad( A j )) = ker (cid:0) End C ( C r ) Tr −→ C (cid:1) . (cid:3) For the relative connection(47) ∇ P : (cid:0) O hol P ×V (cid:1) ⊕ r −→ (cid:0) O hol P ×V (cid:1) ⊕ r ⊗ Ω P ×V / V (cid:0) D V ∪ ( {∞} × V ) (cid:1) hol constructed in Proposition 4.3, let A ∞ ( z, (cid:15) ) dzz m − (cid:15) m be the connection matrix of ∇ P . Since ∇ P is regularsingular at z = ∞ , we can write A ∞ ( z, (cid:15) ) = A ∞ , ( (cid:15) ) + A ∞ , ( (cid:15) ) z + · · · + A ∞ ,m − ( (cid:15) ) z m − with matrices A ∞ , ( (cid:15) ) , . . . , A ∞ ,m − ( (cid:15) ) of holomorphic functions in ( (cid:15), w ) ∈ V . Using ∇ P | ∆ ×V = ∇ ∆ | ∆ ×V and (46), we can see that there exists an invertible matrix P ( z, (cid:15) ) of holomorphic functions on a neighbor-hood of D V such that(48) (cid:18) P ( z, (cid:15) ) − dP ( z, (cid:15) ) + P ( z, (cid:15) ) − A ∞ ( z, (cid:15) ) dzz m − (cid:15) m P ( z, (cid:15) ) (cid:19) (cid:12)(cid:12)(cid:12) D V = Diag ( ν ( µ k )) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V . Since ν ( µ ) | p , . . . , ν ( µ r ) | p are distinct at any point p ∈ D V , there exists a polynomial ¯ ψ ( T ) = ¯ a r − T r − + · · · + ¯ a T + ¯ a ∈ O holD V [ T ] satisfying¯ ψ (cid:16) Diag ( ν ( µ k )) (cid:17) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V = Diag ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V . After shrinking V , we can take lifts a ( z, (cid:15) ) , a ( z, (cid:15) ) , . . . , a r − ( z, (cid:15) ) ∈ O hol V [ z ] of ¯ a , ¯ a , . . . , ¯ a r − and put ψ ( T ) := a r − ( z, (cid:15) ) T r − + a r − ( z, (cid:15) ) T r − + · · · + a ( z, (cid:15) ) T + a ( z, (cid:15) ) ∈ O V [ z ][ T ] . Here we may assume that a ( z, (cid:15) ) , . . . , a r − ( z, (cid:15) ) are polynomials in z of degree less than m . Then ψ ( A ∞ ( z, (cid:15) )) is a matrix of polynomials in z and we have P ( z, (cid:15) ) − ψ ( A ∞ ( z, (cid:15) )) P ( z, (cid:15) ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V = Diag ( µ k ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V . For l = 0 , , . . . , r − j (cid:48) = 0 , , . . . , m − 2, we have res z = ∞ (cid:32) Tr (cid:32) ψ ( A ∞ ( z, (cid:15) )) l z j (cid:48) dzz m − (cid:15) m (cid:33)(cid:33) = − m (cid:88) j =1 res z = (cid:15)ζ jm (cid:32) Tr (cid:32) ψ ( A ∞ ( z, (cid:15) )) l z j (cid:48) dzz m − (cid:15) m (cid:33)(cid:33) = − m (cid:88) j =1 res z = (cid:15)ζ jm (cid:32) Tr (cid:32) P ( z, (cid:15) ) − ψ ( A ∞ ( z, (cid:15) )) l P ( z, (cid:15) ) z j (cid:48) dzz m − (cid:15) m (cid:33)(cid:33) = − m (cid:88) j =1 res z = (cid:15)ζ jm (cid:32) Tr (cid:32)(cid:16) Diag ( µ k ) (cid:17) l z j (cid:48) dzz m − (cid:15) m (cid:33)(cid:33) = res z = ∞ (cid:32) Tr (cid:32) Diag ( µ lk ) z j (cid:48) dzz m − (cid:15) m (cid:33)(cid:33) = 0 . We can write ψ ( A ∞ ( z, (cid:15) )) l = Q (cid:88) q =0 C ( l ) q ( (cid:15) ) z q NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 47 for matrices C ( l ) q ( (cid:15) ) constant in z . We define(49) Ξ l,j ( z, (cid:15) ) := m − (cid:88) j (cid:48) =0 (cid:88) p ≥ ≤ pm + j (cid:48)− j ≤ Q (cid:15) pm z j (cid:48) C ( l ) pm + j (cid:48) − j ( (cid:15) )for j = 0 , , . . . , m − l = 0 , . . . , r − 1. In other words, Ξ l,j ( z, (cid:15) ) is obtained from z j ψ ( A ∞ ( z, (cid:15) )) l bysubstituting (cid:15) m in z m . Then we have A ∞ ( z, (cid:15) ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V = P ( z, (cid:15) ) ν (cid:16) Diag ( µ k ) (cid:17) P ( z, (cid:15) ) − dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V = r − (cid:88) l =0 m − (cid:88) j =0 c l,j z j ψ ( A ∞ ( z, (cid:15) )) l dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V = r − (cid:88) l =0 m − (cid:88) j =0 c l,j Ξ l,j ( z, (cid:15) ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V , from which we have A ∞ ( z, (cid:15) ) = r − (cid:88) l =0 m − (cid:88) j =0 c l,j Ξ l,j ( z, (cid:15) ) . Note that we have res z = ∞ (cid:18) Tr (cid:18) Ξ l,j ( z, (cid:15) ) dzz m − (cid:15) m (cid:19)(cid:19) = − Tr (cid:16) (cid:88) ≤ pm + m − − j ≤ Q (cid:15) pm C ( l ) pm + m − − j ( (cid:15) ) (cid:17) (50) = res z = ∞ (cid:18) Tr (cid:18) z j ψ ( A ∞ ( z, (cid:15) )) l dzz m − (cid:15) m (cid:19)(cid:19) = 0for j = 0 , , . . . , m − V (cid:15) m := V × ∆ × ∆ s (Spec C [ (cid:15) ] / ( (cid:15) m ) × ∆ s ) and V (cid:15) m [¯ h ] := V (cid:15) m × Spec C [ h ] / ( h ). Then the restriction ∇ P | P ×V (cid:15)m : ( O hol P ×V (cid:15)m ) ⊕ r −→ ( O hol P ×V (cid:15)m ) ⊕ r ⊗ Ω P ×V (cid:15)m / V (cid:15)m ( D V (cid:15)m ∪ ( ∞ × V (cid:15) m ))(51) f ...f r (cid:55)→ df ...df r + A ∞ ( z, ¯ (cid:15) ) dzz m f ...f r of the relative connection ∇ P given in (47) to P × V (cid:15) m becomes a relative irregular singular connection,where A ∞ ( z, ¯ (cid:15) ) is the restriction of A ∞ ( z, (cid:15) ) to P × V (cid:15) m . If we put B ,l,j ( z ) := P ( z, ¯ (cid:15) ) Diag (cid:0) (cid:82) µ lk z j dzzm (cid:1) P ( z, ¯ (cid:15) ) − for j = 0 , , . . . , m − l = 0 , , . . . , r − 1, then B ,l,j ( z ) becomes a matrix of single valued meromorphicforms whose pole order at z = 0 is at most m − 1, because µ lk z j dzz m has no residue part. If we put(52) A (cid:15) m , ¯ h,v l,j ( z ) dzz m := dB ,l,j ( z ) + [ A ∞ ( z, ¯ (cid:15) ) , B ,l,j ( z )] dzz m , then we can see that P ( z, ¯ (cid:15) ) − A (cid:15) m , ¯ h,v l,j ( z ) P ( z, ¯ (cid:15) ) (cid:12)(cid:12) D V (cid:15)m = Diag ( µ lk z j ) (cid:12)(cid:12) D V (cid:15)m because of (48). Let us considerthe connection ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j : ( O hol ∆ ×V (cid:15)m [¯ h ] ) ⊕ r −→ ( O hol ∆ ×V (cid:15)m [¯ h ] ) ⊕ r ⊗ Ω ∆ ×V (cid:15)m [¯ h ] / V (cid:15)m ( D V (cid:15)m [¯ h ] )(53) f ...f r (cid:55)→ df ...df r + (cid:18) ( A ∞ ( z, ¯ (cid:15) ) + ¯ hA (cid:15) m , ¯ h,v l,j ( z )) dzz m + B ,l,j ( z ) d ¯ h (cid:19) f ...f r . Lemma 4.5. The connection ∇ flat ∆ ×V (cid:15) [¯ h ] ,v l,j given in (53) satisfies the integrability condition d (cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19) + (cid:20)(cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19) , (cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19)(cid:21) = 0 . Proof. The lemma follows from the immediate calculation d (cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19) + (cid:20)(cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19) , (cid:18) ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m + B ,l,j d ¯ h (cid:19)(cid:21) = d ¯ h ∧ A (cid:15) m , ¯ h,v l,j dzz m + dB ,l,j ∧ d ¯ h + (cid:20) A ∞ dzz m , B ,l,j d ¯ h (cid:21) = 0using (52). (cid:3) We choose a fundamental solution Y , ∞ ( z ) of ∇ P ∆ ×V (cid:15)m and put ˜ Y , ∞ ( z, ¯ h ) := Y , ∞ ( z ) − ¯ hB ,l,j ( z ) Y , ∞ ( z ). Lemma 4.6. ˜ Y , ∞ ( z, ¯ h ) = Y , ∞ ( z ) − ¯ hB ,l,j ( z ) Y , ∞ ( z ) is a fundamental solution of the relative connection (54) ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j : ( O hol ∆ ×V (cid:15)m [¯ h ] ) ⊕ r −→ ( O hol ∆ ×V (cid:15)m [¯ h ] ) ⊕ r ⊗ Ω ∆ ×V (cid:15)m [¯ h ] / V (cid:15)m [¯ h ] ( D V (cid:15)m [¯ h ] ) induced by ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j , whose connection matrix is ( A ∞ + ¯ hA (cid:15) m , ¯ h,v l,j ) dzz m .Proof. The lemma follows from the calculation ∂∂z (cid:0) Y , ∞ − ¯ hB ,l,j ( z ) Y , ∞ (cid:1) dz = dY , ∞ ( z ) − ¯ h (cid:0) dB ,l,j ( z ) Y , ∞ + B ,l,j ( z ) dY , ∞ (cid:1) (55) = − A ∞ ( z, ¯ (cid:15) ) dzz m Y , ∞ − ¯ hA (cid:15) m , ¯ h,v l,j ( z ) dzz m Y , ∞ + ¯ h (cid:16)(cid:2) A ∞ ( z, ¯ (cid:15) ) , B ,l,j ( z ) (cid:3) + B ,l,j ( z ) A ∞ ( z, ¯ (cid:15) ) (cid:17) dzz m Y , ∞ = − (cid:0) A ∞ ( z, ¯ (cid:15) ) + ¯ hA (cid:15) m , ¯ h,v l,j ( z ) (cid:1) dzz m (cid:0) Y , ∞ − ¯ hB ,l,j ( z ) Y , ∞ (cid:1) . (cid:3) Let Mon ˜ γ be the monodromy matrix of Y , ∞ ( z ) along ˜ γ . Then ˜ Y , ∞ ( z, ¯ h ) = Y , ∞ ( z ) − ¯ hB ,l,j ( z ) Y , ∞ ( z )has the monodromy matrix Mon ˜ γ along ˜ γ , because B ,l,j ( z ) is single valued on (∆ × V (cid:15) m ) \ D V (cid:15)m . By thesimilar method to that in the proof of Proposition 4.3, we can construct a global connection ∇ P ×V (cid:15)m [¯ h ] ,v l,j : ( O hol P ×V (cid:15)m [¯ h ] ) ⊕ r −→ ( O hol P ×V (cid:15)m [¯ h ] ) ⊕ r ⊗ Ω P ×V (cid:15)m [¯ h ] / V [¯ h ] (cid:16) D V (cid:15)m [¯ h ] ∪ ( ∞ × V (cid:15) m [¯ h ]) (cid:17) f ...f r (cid:55)→ df ...df r + ( A ∞ ( z, ¯ (cid:15) ) + ¯ h ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z )) dzz m − (cid:15) m f ...f r satisfying res z = ∞ (cid:18) ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z ) dzz m (cid:19) = 0such that the restriction of ∇ P ×V (cid:15)m [¯ h ] ,v l,j to P × V (cid:15) m coincides with the restriction ∇ P | P ×V (cid:15)m given in(51) and that the restriction of ∇ P ×V (cid:15)m [¯ h ] ,v l,j to ∆ × V (cid:15) m [¯ h ] is isomorphic to the irregular singular relativeconnection ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j given in (54). By construction, there is a convergent power series ∞ (cid:88) l (cid:48) =0 R (cid:48) ( l )0 ,j,l (cid:48) z l (cid:48) such that( A ∞ ( z, ¯ (cid:15) ) + ¯ hA (cid:15) m , ¯ h,v l,j ( z )) dzz m = ¯ h ∞ (cid:88) l (cid:48) =1 l (cid:48) R (cid:48) ( l )0 ,j,l (cid:48) z l (cid:48) − dz + (cid:16) − ¯ h ∞ (cid:88) l (cid:48) =0 R (cid:48) ( l )0 ,j,l (cid:48) z l (cid:48) (cid:17)(cid:0) A ∞ ( z, ¯ (cid:15) ) + ¯ h ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z ) (cid:1) dzz m (cid:16) h ∞ (cid:88) l (cid:48) =0 R (cid:48) ( l )0 ,j,l (cid:48) z l (cid:48) (cid:17) , NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 49 which impliesΞ l,j ( z, ¯ (cid:15) ) (cid:12)(cid:12) D V (cid:15)m = ψ ( A ∞ ( z, ¯ (cid:15) )) l z j (cid:12)(cid:12) D V (cid:15)m = P ( z, ¯ (cid:15) ) Diag ( µ lk z j ) P ( z, ¯ (cid:15) ) − (cid:12)(cid:12) D V (cid:15)m = A (cid:15) m , ¯ h,v l,j ( z ) (cid:12)(cid:12) D V (cid:15)m = (cid:18) ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z ) + m − (cid:88) j (cid:48) =0 j (cid:48) (cid:88) l (cid:48) =0 (cid:104) A ∞ ,j (cid:48) − l (cid:48) (¯ (cid:15) ) , R (cid:48) ( l )0 ,j,l (cid:48) (cid:105) z j (cid:48) (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) D V (cid:15)m . So we have(56) Ξ l,j ( z, ¯ (cid:15) ) = ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z ) + m − (cid:88) j (cid:48) =0 j (cid:48) (cid:88) l (cid:48) =0 (cid:104) A ∞ ,j (cid:48) − l (cid:48) (¯ (cid:15) ) , R (cid:48) ( l )0 ,j,l (cid:48) (cid:105) z j (cid:48) . We put B (cid:48) ,l,j ( z ) := B ,l,j ( z ) − ∞ (cid:88) l (cid:48) =0 R (cid:48) ( l )0 ,j,l (cid:48) z l (cid:48) . Lemma 4.7. The connection on ( O hol ∆ ×V (cid:15)m [¯ h ] ) ⊕ r given by the connection matrix ( A ∞ ( z, ¯ (cid:15) ) + ¯ h ˜ A (cid:48) (cid:15) m , ¯ h,v l,j ( z )) dzz m + B (cid:48) ,l,j ( z ) d ¯ h is isomorphic to the connection ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j given in (53) and satisfies the integrability condition.Proof. Indeed the isomorphism is given by I r + ¯ h (cid:80) ∞ l (cid:48) =0 B (cid:48) ,j,l (cid:48) z l (cid:48) and the integrability follows from that of ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j . (cid:3) We will give a lift of the connection given in Lemma 4.7 as a connection on ∆ × V [¯ h ], by means ofextending the data ( R (cid:48) ( l )0 ,j,l (cid:48) ). Definition 4.8. We say that (cid:0) R ( l ) j,l (cid:48) ( (cid:15) ) (cid:1) ≤ l ≤ r − ≤ j ≤ m − , ≤ l (cid:48) ≤ r − is an adjusting data for the connection ∇ P givenin (47) if each R ( l ) j,l (cid:48) ( (cid:15) ) is a matrix whose entries belong to O hol V such that R ( l ) j,l (cid:48) ( (cid:15) ) (cid:12)(cid:12) (cid:15) m =0 = R (cid:48) ( l )0 ,j,l (cid:48) and thatthe z m − -coefficient of Ξ l,j ( z, (cid:15) ) given in (49) is expressed by(57) (cid:88) ≤ pm + m − − j ≤ Q (cid:15) pm C ( l ) pm + m − − j ( (cid:15) ) = m − (cid:88) l (cid:48) =0 (cid:104) A ∞ ,m − l (cid:48) − ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) . Lemma 4.9. There exists an adjusting data (cid:0) R ( l ) j,l (cid:48) ( (cid:15) ) (cid:1) ≤ l ≤ r − ≤ j ≤ m − , ≤ l (cid:48) ≤ r − for the connection ∇ P .Proof. For each u ∈ (cid:84) m − j =0 ker(ad( A ∞ ,j ( (cid:15) ))), we have u · A ∞ ( z, (cid:15) ) dzz m − (cid:15) m − A ∞ ( z, (cid:15) ) dzz m − (cid:15) m · u = 0. So u | ∆ ×{ b } is a section of ker ∇ † ∆ b on ∆ × { b } for each b ∈ V , which is a scalar endomorphism by Assumption4.2, (ii). Then we have u ∈ O hol V · id and(58) m − (cid:92) j =0 ker (ad( A ∞ ,j ( (cid:15) ))) = O hol V · idfollows. So we can see m − (cid:88) j =0 im(ad( A ∞ ,j ( (cid:15) ))) = ker (cid:16) E nd O hol V (cid:16)(cid:0) O hol V (cid:1) ⊕ r (cid:17) Tr −→ O hol V (cid:17) , because the equality for the restriction to each b (cid:48) ∈ V holds by Lemma 4.4. Then, after shrinking V , thereare matrices R ( l ) j, ( (cid:15) ) , . . . , R ( l ) j,m − ( (cid:15) ) constant in z such that (cid:88) ≤ pm + m − − j ≤ Q (cid:15) pm C ( l ) pm + m − − j ( (cid:15) ) = m − (cid:88) l (cid:48) =0 (cid:104) A ∞ ,m − l (cid:48) − ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) . because of (50). Here we may assume R ( l ) j,l (cid:48) ( (cid:15) ) (cid:12)(cid:12) (cid:15) m =0 = R (cid:48) ( l )0 ,j,l (cid:48) by using (56). (cid:3) For l = 0 , , . . . , r − j = 0 , , . . . , m − 2, we take an adjusting data (cid:0) R ( l ) j,l (cid:48) ( (cid:15) ) (cid:1) ≤ l ≤ r − ≤ j ≤ m − , ≤ l (cid:48) ≤ r − for the connection ∇ P and define˜Ξ l,j ( z, (cid:15) ) := Ξ l,j ( z, (cid:15) ) − m − (cid:88) q =0 (cid:88) ≤ l (cid:48) ≤ m − − q (cid:104) A ∞ ,q ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) z q + l (cid:48) (59) − m − (cid:88) q =0 (cid:88) m − q ≤ l (cid:48) ≤ m − (cid:104) A ∞ ,q ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) (cid:15) m z q + l (cid:48) − m . Then, using (57), we have the equality res z = ∞ (cid:18) ˜Ξ l,j ( z, (cid:15) ) dzz m − (cid:15) m (cid:19) = res z = ∞ (cid:32) Ξ l,j ( z, (cid:15) ) dzz m − (cid:15) m − m − (cid:88) l (cid:48) =0 (cid:104) A ∞ ,m − l (cid:48) − ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) z m − dzz m − (cid:15) m (cid:33) (60) = − (cid:88) ≤ pm + m − − j ≤ Q (cid:15) pm C ( l ) pm + m − − j ( (cid:15) ) + m − (cid:88) l (cid:48) =0 (cid:104) A ∞ ,m − l (cid:48) − ( (cid:15) ) , R ( l ) j,l (cid:48) ( (cid:15) ) (cid:105) = 0for j = 0 , , . . . , m − l = 0 , , . . . , r − l,j ( z, (cid:15) ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12)(cid:12) D V = P ( z, (cid:15) ) z j Diag ( µ lk ) P ( z, (cid:15) ) − dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V − (cid:104) A ∞ ( z, (cid:15) ) , m − (cid:88) l (cid:48) =0 R ( l ) j,l (cid:48) ( (cid:15) ) z l (cid:48) (cid:105) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12) D V . Let(61) ∇ P ×V ,v l,j : ( O hol P ×V [¯ h ] ) ⊕ r −→ ( O hol P ×V [¯ h ] ) ⊕ r ⊗ Ω P ×V [¯ h ] / V [¯ h ] (cid:16) D V [¯ h ] ∪ ( ∞ × V [¯ h ]) (cid:17) hol be the relative connection defined by ∇ P ×V [¯ h ] ,v l,j f ...f r = df ...df r + (cid:16) A ∞ ( z, (cid:15) ) + ¯ h ˜Ξ l,j ( z, (cid:15) ) (cid:17) dzz m − (cid:15) m f ...f r . Then ∇ P ×V [¯ h ] ,v l,j (cid:12)(cid:12) ∆ ×V (cid:15)m [¯ h ] is isomorphic to ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j by the construction. Using (60), we can see theequality res z = ∞ ( ∇ P ×V [¯ h ] ,v l,j ) = res z = ∞ (cid:0) ∇ P (cid:1) . By construction, there is an invertible matrix ˜ P ( z, ¯ h ) suchthat (cid:16) ˜ P ( z, ¯ h ) − d ˜ P ( z, ¯ h ) + ˜ P ( z, ¯ h ) − (cid:16) A ∞ ( z, (cid:15) ) + ¯ h ˜Ξ l,j ( z, (cid:15) ) (cid:17) dzz m − (cid:15) m ˜ P ( z, ¯ h ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) D V [¯ h ] = Diag ( ν ( µ k )+¯ hµ lk z j ) dzz m − (cid:15) m (cid:12)(cid:12)(cid:12)(cid:12) D V [¯ h ] . We may further assume that ˜ P ( z, ¯ h ) P ( z ) − (cid:12)(cid:12) D V [¯ h ] = (cid:16) I r + ¯ h m − (cid:88) l (cid:48) =0 R ( l ) j,l (cid:48) z l (cid:48) (cid:17)(cid:12)(cid:12)(cid:12) D V [¯ h ] . We will construct an integrable connection on P × V [¯ h ] over V which is an extension of (61). Definition 4.10. We say that a connection ∇ flat P ×V [¯ h ] ,v l,j : ( O hol P × ˜ V [¯ h ] ) ⊕ r −→ ( O hol P × ˜ V [¯ h ] ) ⊕ r ⊗ ( ι V [¯ h ] ) ∗ Ω P ×V\ Γ V )[¯ h ] (cid:14) V (cid:0) ∞ × V [¯ h ] (cid:1) hol (62) f ...f r (cid:55)→ df ...df r + (cid:18) ( A ∞ ( z, (cid:15) ) + ¯ h ˜Ξ l,j ( z, (cid:15) )) dzz m − (cid:15) m + B l,j ( z, (cid:15) ) d ¯ h (cid:19) f ...f r NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 51 is a horizontal lift of ∇ P ×V ,v l,j if B l,j ( z, (cid:15) ) | (cid:15) m =0 = B (cid:48) ,l,j ( z ) and ∇ flat P ×V [¯ h ] ,v l,j is integrable in the sense that d (cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19) + (cid:20)(cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19) , (cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19)(cid:21) = 0 . Proposition 4.11. There exists a horizontal lift ∇ flat P ×V [¯ h ] ,v l,j : ( O hol P × ˜ V [¯ h ] ) ⊕ r −→ ( O hol P × ˜ V [¯ h ] ) ⊕ r ⊗ ( ι V [¯ h ] ) ∗ Ω P ×V\ Γ V )[¯ h ] (cid:14) V (cid:0) ∞ × V [¯ h ] (cid:1) hol of the relative connection ∇ P ×V ,v l,j given in (61) after shrinking V , where ι V [¯ h ] : ( P ×V \ Γ V )[¯ h ] (cid:44) → P ×V [¯ h ]is the canonical inclusion.. Proof. After shrinking V , we can take a local basis ˜ Y ∞ ( z, (cid:15), ¯ h ) of ker( ∇ P ×V [¯ h ] ,v l,j ) on ( U ∞ \ Γ ∞ ) × V [¯ h ]for some open neighborhood U ∞ of ∞ in P and a slit Γ ∞ ⊂ U ∞ which is a simple path joining ∞ anda boundary point b ∞ ∈ ∂U ∞ of U ∞ . Here we may assume that the restriction Y ∞ ( z, ¯ (cid:15) ) of ˜ Y ∞ ( z, (cid:15), ¯ h )to ( U ∞ \ Γ ∞ ) × V (cid:15) m coincides with Y , ∞ ( z ) which is chosen before Lemma 4.6. We may further assumethat the monodromy matrix Mon ∞ ( (cid:15) ) of ˜ Y ∞ ( z, (cid:15), ¯ h ) around ∞ × V [¯ h ] coincides with that of Y ∞ ( z, (cid:15) ) :=˜ Y ∞ ( z, (cid:15), ∇ P ×V [¯ h ] ,v l,j at z = ∞ is constant in ¯ h .Consider the restriction ˜ Y ∞ ( z, ¯ (cid:15), ¯ h ) of ˜ Y ∞ ( z, (cid:15), ¯ h ) to ( U ∞ \ Γ ∞ ) × V (cid:15) m [¯ h ]. Using the integrability conditionof ∇ flat ∆ ×V (cid:15)m [¯ h ] ,v l,j , we can see in the same way as (55) that Y ∞ ( z, ¯ (cid:15) ) − ¯ hB (cid:48) ,l,j ( z ) Y ∞ ( z, ¯ (cid:15) ) is a fundamentalsolution of ∇ P ×V [¯ h ] (cid:12)(cid:12) ( U ∞ \ Γ ∞ ) ×V (cid:15)m [¯ h ] after an analytic continuation. So we can write Y ∞ ( z, ¯ (cid:15) ) − ¯ hB (cid:48) ,l,j ( z ) Y ∞ ( z, ¯ (cid:15) ) = ˜ Y ∞ ( z, ¯ (cid:15), ¯ h ) C (¯ (cid:15), ¯ h )for a matrix C (¯ (cid:15), ¯ h ) constant in z . Since both Y ∞ ( z, ¯ (cid:15) ) − ¯ hB (cid:48) ,l,j ( z ) Y ∞ ( z, ¯ (cid:15) ) and ˜ Y ∞ ( z, ¯ (cid:15), ¯ h ) have the samemonodromy Mon ∞ (¯ (cid:15) ) := Mon ∞ ( (cid:15) ) | (cid:15) m =0 , we should have (cid:0) Y ∞ ( z, ¯ (cid:15) ) − ¯ hB (cid:48) ,l,j ( z ) Y ∞ ( z, ¯ (cid:15) ) (cid:1) Mon ∞ (¯ (cid:15) ) = ˜ Y ∞ ( z, ¯ (cid:15), ¯ h ) Mon ∞ (¯ (cid:15) ) C (¯ (cid:15), ¯ h )from which we have C (¯ (cid:15), ¯ h ) Mon ∞ (¯ (cid:15) ) = Mon ∞ (¯ (cid:15) ) C (¯ (cid:15), ¯ h ) . So we can write C (¯ (cid:15), ¯ h ) = r − (cid:88) l =0 b l (¯ (cid:15), ¯ h )Mon ∞ (¯ (cid:15) ) l , because Mon ∞ (¯ (cid:15) ) | b has the r distinct eigenvalues at each b ∈ V (cid:15) m . Shrinking V , we can take lifts b l ( (cid:15), ¯ h ) of b l (¯ (cid:15), ¯ h ) as holomorphic functions in (cid:15) . If we replace ˜ Y ∞ ( z, (cid:15), ¯ h ) by ˜ Y ∞ ( z, (cid:15), ¯ h ) (cid:80) r − l =0 b l ( (cid:15), ¯ h )Mon ∞ ( (cid:15) ) l , thenthe restriction of ˜ Y ∞ ( z, (cid:15), ¯ h ) to ( U ∞ × V (cid:15) m [¯ h ]) \ (Γ ∞ × V (cid:15) m [¯ h ]) coincides with Y ∞ ( z, ¯ (cid:15) ) − ¯ hB (cid:48) ,l,j ( z ) Y ∞ ( z, ¯ (cid:15) ).If we define(63) B l,j ( z, (cid:15) ) := − ∂ ˜ Y ∞ ( z, (cid:15), ¯ h ) ∂ ¯ h Y ∞ ( z, (cid:15) ) − , we have B l,j ( z, (cid:15) ) | (cid:15) m =0 = B (cid:48) ,l,j ( z ). Since both ˜ Y ∞ ( z, (cid:15), ¯ h ) and Y ∞ ( z, (cid:15) ) have the same monodromy matrixaround ∞ , we can regard B l,j ( z, (cid:15) ) as a matrix of single valued holomorphic functions on ( P × V ) \ Γ V after an analytic continuation. Let us consider the connection ∇ flat P ×V [¯ h ] ,v l,j : ( O hol P × ˜ V [¯ h ] ) ⊕ r −→ ( O hol P × ˜ V [¯ h ] ) ⊕ r ⊗ ( ι V [¯ h ] ) ∗ Ω P ×V\ Γ V )[¯ h ] (cid:14) V (cid:0) ∞ × V [¯ h ] (cid:1) hol f ...f r (cid:55)→ df ...df r + (cid:18) ( A ∞ ( z, (cid:15) ) + ¯ h ˜Ξ l,j ( z, (cid:15) )) dzz m − (cid:15) m + B l,j ( z, (cid:15) ) d ¯ h (cid:19) f ...f r . The curvature form of ∇ flat P ×V [¯ h ] ,v l,j becomes d (cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19) + (cid:20)(cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19) , (cid:18)(cid:0) A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) (cid:1) dzz m − (cid:15) m + B l,j ( z ) d ¯ h (cid:19)(cid:21) = ˜Ξ l,j ( z ) d ¯ h ∧ dzz m − (cid:15) m + ∂B l,j ( z ) ∂z dz ∧ d ¯ h + ( A ∞ ( z ) B l,j ( z ) − B l,j ( z ) A ∞ ( z )) dzz m − (cid:15) ∧ d ¯ h = − ˜Ξ l,j ( z ) dzz m − (cid:15) m ∧ d ¯ h − ∂∂z (cid:16) ∂ ˜ Y ∞ ∂ ¯ h ( z, ¯ h ) Y ∞ ( z ) − (cid:17) dz ∧ d ¯ h + (cid:16) − A ∞ ( z ) ∂ ˜ Y ∞ ∂ ¯ h ( z, ¯ h ) Y ∞ ( z ) − + ∂ ˜ Y ∞ ∂ ¯ h ( z, ¯ h ) Y ∞ ( z ) − A ∞ ( z ) (cid:17) dzz m − (cid:15) m ∧ d ¯ h = − ˜Ξ l,j ( z ) dzz m − (cid:15) m ∧ d ¯ h − (cid:18) ∂ ˜ Y ∞ ∂ ¯ h∂z Y − ∞ (cid:19) dz ∧ d ¯ h + (cid:18) ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ ∂Y ∞ ∂z Y − ∞ (cid:19) dz ∧ d ¯ h − (cid:16) A ∞ ( z ) ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ − ∂ ˜ Y ϑ ∂ ¯ h Y − ∞ A ∞ ( z ) (cid:17) dzz m − (cid:15) m ∧ d ¯ h = − ˜Ξ l,j ( z ) dzz m − (cid:15) m ∧ d ¯ h − ∂∂ ¯ h (cid:18) − A ∞ ( z ) + ¯ h ˜Ξ l,j ( z ) z m − (cid:15) m ˜ Y ∞ (cid:19) Y − ∞ dz ∧ d ¯ h − ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ A ∞ ( z ) dzz m − (cid:15) m ∧ d ¯ h − (cid:16) A ∞ ( z ) ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ − ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ A ∞ ( z ) (cid:17) dzz m − (cid:15) m ∧ d ¯ h = − ˜Ξ l,j ( z ) dzz m − (cid:15) m ∧ d ¯ h + ˜Ξ l,j ( z ) z m − (cid:15) m dz ∧ d ¯ h + A ∞ ( z ) z m − (cid:15) m ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ dz ∧ d ¯ h − ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ A ∞ ( z ) dzz m − (cid:15) m ∧ d ¯ h − A ∞ ( z ) ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ dzz m − (cid:15) m ∧ d ¯ h + ∂ ˜ Y ∞ ∂ ¯ h Y − ∞ A ∞ ( z ) dzz m − (cid:15) m ∧ d ¯ h = 0 . So ∇ flat P ×V [¯ h ] ,v l,j is an integrable connection and becomes a horizontal lift of ∇ P ×V ,v l,j . (cid:3) Comparison with the asymptotic property in the unfolding theory by Hurtubise, Lambertand Rousseau. In the unfolding theory by Hurtubise, Lambert and Rousseau in [14], [15], unfolded Stokesmatrices for unfolded linear differential equations are defined. So our integrable connection ∇ flat P ×V [¯ h ] ,v l,j constructed in Proposition 4.11 induces unfolded Stokes matrices but we cannot expect that these matricesare constant in ¯ h . Although we cannot produce any positive result on the asymptotic property concernedwith the integrable connection ∇ flat P ×V [¯ h ] ,v l,j defined by (62) in subsection 4.2, it will be worth pointing outwhat is the difficulty.We use the same notations as in subsection 4.1 and in subsection 4.2. We consider the multivaluedfunction τ (cid:15) ( z ) := (cid:90) dzz m − (cid:15) m which is single valued on P × ∆ s \ Γ ∆ . Under a suitable choice of path integral, we may assume that τ (cid:15) ( z )does not vanish on Γ ∆ \ (Γ ∆ ∩ D ). Let (cid:36) : [0 , × S −→ ∆( s, e √− ψ ) (cid:55)→ se √− ψ be the polar blow up. We can regard ∆ × [0 , × S × ∆ s ⊂ C × [0 , × S × ∆ s ⊂ P × [0 , × S × ∆ s .By Proposition 3.1, we can take an open neighborhood U of { } × { } × S × ∆ s in ∆ × [0 , × S × ∆ s and an open covering U \ ((id × (cid:36) × id) − ( D ) ∩ U )) = m (cid:91) j =1 (cid:91) ≤ ψ ≤ π (cid:91) ξ =1 W ( j ) ψ ,ξ NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 53 such that any flow of the vector field v (cid:15),θ = Re (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂x + Im (cid:16) e √− θ ( z m − (cid:15) m ) (cid:17) ∂∂y starting at a point in W ( j ) ψ ,ξ has an accumulation point in (id × (cid:36) − × id) − ( D ) ∩ U , where x = Re( z ), y = Im( z ). Here θ = θ ( j ) ψ ,ξ ∈ R is determined by j, ψ , ξ as in the proof of Proposition 3.1.We take an open covering ( (cid:36) × id ∆ s ) − ( V ) = (cid:91) b ∈ ( (cid:36) × id ∆ s ) − ( V ) ˜ V (cid:48) b by small contractible open subsets V (cid:48) b of ( (cid:36) × id ∆ s ) − ( V ). By Theorem 3.2, we can see that there are anopen covering (∆ × ˜ V (cid:48) b ) ∩ W ( j ) ψ ,ξ = (cid:91) p ∈ W ( j ) ψ ,ξ S ( j ) ψ ,ξ,p = (cid:91) ϑ S ϑ with ϑ = ( j, ψ , ξ, p ) and a matrix Y ϑ ( z, s, e √− ψ , w, h ) = (cid:16) ˜ y ϑ (cid:0) z, s, e √− ψ , w, h (cid:1) , . . . , ˜ y ϑr (cid:0) z, s, e √− ψ , w, h (cid:1)(cid:17) of functions on S ϑ × ∆ δ for some δ > 0, satisfying(64) d ˜ Y ϑ ( z, s, e √− ψ , w, h ) dz = − A ∞ ( z, (cid:15), w ) + h ˜Ξ ( l ) q ( z, (cid:15), w ) z m − (cid:15) m ˜ Y ϑ ( z, s, e √− ψ , w, h ) , such that the limit(65) lim t →∞ ˜ P ( z θ ( t ) , h ) ˜ Y ϑ ( z θ ( t ) , h ) Diag (cid:0) exp (cid:0) (cid:82) tt ( ν ( µ k )( z θ ( t ))+ hµ lk z θ ( t ) q ) e √− θ dt (cid:1)(cid:1) = I r is the identity matrix, where z θ ( t ) is a flow of v (cid:15),θ in S ϑ = S ( j ) ψ ,ξ,p and θ = θ ( j ) ψ ,ξ is determined from ϑ =( j, ψ , ξ, p ). We denote the restriction of ˜ Y ϑ ( z, s, e √− ψ , h ) to S ϑ [¯ h ] by ˜ Y ϑ ( z, ¯ h ) and denote the restrictionof ˜ Y ϑ ( z, s, e √− ψ , h ) to S ϑ × { } by Y ϑ ( z ). By (65), we have(66) lim t →∞ ˜ P ( z θ ( t ) , ¯ h ) ˜ Y ϑ ( z θ ( t ) , ¯ h ) Diag (cid:0) exp (cid:0) (cid:82) tt ( ν ( µ k )( z θ ( t ))) e √− θ dt (cid:1)(cid:0) h (cid:82) tt µ lk z θ ( t ) l e √− θ dt ) (cid:1)(cid:1) = I r from which ˜ Y ϑ ( z, ¯ h )Diag (exp( (cid:82) ν ( µ k )( z ) dzzm − (cid:15)m )) (cid:0) I r +¯ h Diag ( µ lk z q dzzm − (cid:15)m ) (cid:1) is bounded on S ϑ [¯ h ] and in particular Y ϑ ( z )Diag (exp( (cid:82) ν ( µ k )( z ) dzzm − (cid:15)m )) is bounded on S ϑ .Recall that we can write Y ∞ ( z ) = (cid:0) y ∞ ( z ) , . . . , y ∞ r ( z ) (cid:1) for y ∞ k ( z ) := ˜ y k ( z, γ : [0 , × ˜ V (cid:48) b −→ (cid:0) ∆ × ˜ V (cid:48) b (cid:1) \ Γ ˜ V (cid:48) b satisfying γ (0 , w ) = γ (1 , w ), p ( γ ( t, w )) = w and that γ ( • , w ) : [0 , −→ ∆ × { w } is homotopic to ˜ γ ( • , w ) for any w ∈ V . From the analysis of flowsin Proposition 3.1, we may assume that there are points 0 = t < t < · · · < t I < t i ∈ S ϑ i ,lim t →∞ z θ i ( t ) = (cid:15)ζ j i m and that either j i +1 = j i + 1 or j i +1 = j i with (cid:15)ζ j i m ∈ S ϑ i ∩ S ϑ i +1 holds. Here in thecase of (cid:15)ζ j i m ∈ S ϑ i ∩ S ϑ i +1 , we can further assume that a flow z θ i ( t ) lie in S ϑ i ∩ S ϑ i +1 which is accumulatedto (cid:15)ζ j i m and a flow z θ i +1 ( t ) lie in S ϑ i ∩ S ϑ i +1 which is accumulated to (cid:15)ζ j i m . Lemma 4.12. Assume that flows z θ ( t ) (resp. z θ (cid:48) ) of v (cid:15),θ (resp. v (cid:15),θ (cid:48) ) in S ϑ (resp. S ϑ (cid:48) ) lie in S ϑ ∩ S ϑ (cid:48) for ϑ, ϑ (cid:48) and that lim t →∞ z θ ( t ) = lim t →∞ z θ (cid:48) ( t ) = (cid:15)ζ jm ∈ S ϑ ∩ S ϑ (cid:48) . We take a permutation σ of { , . . . , r } satisfying Re (cid:0) e √− θ ν ( µ σ (1) )( (cid:15)ζ jm ) (cid:1) > · · · > Re (cid:0) e √− θ ν ( µ σ ( r ) )( (cid:15)ζ jm ) (cid:1) . Assume that ˜ Y ϑ (cid:48) ( z, ¯ h ) = ˜ Y ϑ ( z, ¯ h ) C ϑ,ϑ (cid:48) (¯ h ) holds under an analytic continuation along a path in S ϑ ∪ S ϑ (cid:48) . Then ( e σ (1) , . . . , e σ ( r ) ) − C ϑ,ϑ (cid:48) (¯ h )( e σ (1) , . . . , e σ ( r ) ) is an upper triangular matrix. Proof. We put Λ k ( z, ¯ h ) := exp (cid:18)(cid:90) ( ν ( µ k ) + ¯ hµ lk z q ) dzz m − (cid:15) m (cid:19) . If k < k (cid:48) , then Λ σ ( k ) ( z, ¯ h ) − Λ σ ( k (cid:48) ) ( z, ¯ h ) tends to 0 when z tends to (cid:15)ζ jm . Note that(Diag (Λ k ( z, ¯ h )) ) − C ϑ,ϑ (cid:48) (¯ h ) Diag (Λ k ( z, ¯ h )) = ( ˜ Y ϑ ( z, ¯ h )Diag (Λ k ( z, ¯ h )) ) − ˜ Y ϑ (cid:48) ( z, ¯ h ) Diag (Λ k ( z, ¯ h )) tends to a matrix of bounded functions when z tends to (cid:15)ζ jm in S ϑ ∩ S ϑ (cid:48) .If we put C (cid:48) (¯ h ) := ( e σ (1) , . . . , e σ ( r ) ) − C ϑ,ϑ (cid:48) (¯ h )( e σ (1) , . . . , e σ ( r ) ) = c , (¯ h ) · · · c ,r (¯ h ) ... . . . ...c r, (¯ h ) · · · c r,r (¯ h ) then we have ( e σ (1) , . . . , e σ ( r ) ) − (Diag (Λ k ( z, ¯ h )) ) − C ϑ,ϑ (cid:48) (¯ h )Diag (Λ k ( z, ¯ h )) ( e σ (1) , . . . , e σ ( r ) )= Λ σ (1) · · · ... . . . ... · · · Λ σ ( r ) − c , (¯ h ) · · · c ,r (¯ h ) ... . . . ...c r, (¯ h ) · · · c r,r (¯ h ) Λ σ (1) · · · ... . . . ... · · · Λ σ ( r ) = c , (¯ h ) · · · Λ σ (1) ( z, ¯ h ) − Λ σ ( r ) ( z, ¯ h ) c ,r (¯ h ) ... . . . ... Λ σ (1) ( z, ¯ h )Λ σ ( r ) ( z, ¯ h ) − c r, (¯ h ) · · · c r,r (¯ h ) . Since Λ σ ( k ) ( z, ¯ h ) − Λ σ ( k (cid:48) ) ( z, ¯ h ) is divergent for k > k (cid:48) , we should have c k (cid:48) ,k (¯ h ) = 0 for k (cid:48) > k (cid:3) By an analytic continuation we can write˜ Y ϑ i ( z, ¯ h ) = ˜ Y ∞ ( z, ¯ h ) C ∞ ,ϑ i (¯ h )from which we have ˜ Y ϑ i +1 ( z, ¯ h ) = ˜ Y ϑ i ( z, ¯ h ) C ∞ ,ϑ i (¯ h ) − C ∞ ,ϑ i +1 (¯ h ) . If j i = j i +1 , then ( e σ (1) , . . . , e σ ( r ) ) − C ∞ ,ϑ i (¯ h ) − C ∞ ,ϑ i +1 (¯ h )( e σ (1) , . . . , e σ ( r ) ) is an upper triangular matrixfor a permutation σ by Lemma 4.12. The matrix C ∞ ,ϑ i (¯ h ) is analogous to an unfolded Stokes matrix givenin [14] but we cannot say from its construction that it is constant in ¯ h .We remark that the restriction τ (cid:15) ( z ) − B l,q ( z ) | (cid:15) m =0 = − ( m − z m − B (cid:48) ,l,q ( z ) to the irregular singularlocus (cid:15) m = 0 is bounded around z = 0 by its construction. We can see that B l,q ( z ) = − ∂ ˜ Y ∞ ( z, ¯ h ) ∂ ¯ h Y ∞ ( z ) − = − ∂∂ ¯ h (cid:16) ˜ Y ϑ i ( z, ¯ h ) C ∞ ,ϑ i (¯ h ) − (cid:17) ( ˜ Y ϑ i ( z, C ∞ ,ϑ i (0) − ) − = − ∂ ˜ Y ϑ i ( z, ¯ h ) ∂ ¯ h Y ϑ i ( z ) − + Y ϑ i ( z ) C ∞ ,ϑ i (0) − ∂C ∞ ,ϑ i (¯ h ) ∂ ¯ h Y ϑ i ( z ) − . By the following proposition, we can say that τ (cid:15) ( z ) − ∂ ˜ Y ϑ i ( z, ¯ h ) ∂ ¯ h Y ϑ i ( z ) − is bounded on S ϑ i . However, τ (cid:15) ( z ) − Y ϑ i ( z ) C ∞ ,ϑ i (0) − ∂C ∞ ,ϑ i (¯ h ) ∂ ¯ h Y ϑ i ( z ) − is not bounded unless( e σ (1) , . . . , e σ ( r ) ) − C ∞ ,ϑ i (0) − ∂C ∞ ,ϑ i (¯ h ) ∂ ¯ h ( e σ (1) , . . . , e σ ( r ) )is an upper triangular matrix. So we can not say the boundedness of τ (cid:15) ( z ) − B l,q ( z ) on S ϑ . This is one ofthe reasons why we cannot get a canonical global horizontal lift in section 5. Proposition 4.13. τ (cid:15) ( z ) − ∂∂ ¯ h ˜ Y ϑ ( z, ¯ h ) Y ϑ ( z ) − is bounded on S ϑ . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 55 Proof. Since the limit in (66) is uniform in ¯ h , we can see that T ϑ ( z, ¯ h ) := ˜ Y ϑ ( z, ¯ h ) Diag (cid:0) exp (cid:0) (cid:82) ( ν ( µ k )+¯ hµ lk z q ) dzzm − (cid:15)m (cid:1)(cid:1) and its partial derivative in ¯ h is bounded on ∆ × V [¯ h ]. So ∂T ϑ ( z, ¯ h ) ∂ ¯ h = ∂ ˜ Y ϑ ( z, ¯ h ) ∂ ¯ h Diag (cid:0) exp (cid:0) (cid:82) ν ( µ k ) dzzm − (cid:15)m (cid:1)(cid:1) + Y ϑ ( z ) ∂∂ ¯ h Diag (cid:0) exp (cid:0) (cid:82) ( ν ( µ k )+¯ hµ lk z q ) dzzm − (cid:15)m (cid:1)(cid:1) = ∂ ˜ Y ϑ ( z, ¯ h ) ∂ ¯ h Y ϑ ( z ) − T ϑ ( z ) + T ϑ ( z ) Diag (cid:0) (cid:82) µ lk z q dzzm − (cid:15)m (cid:1) is bounded on S ϑ . So it is sufficient to show that τ (cid:15) ( z ) − Diag (cid:82) ( µ lk z q dzzm − (cid:15)m ) is bounded.If (cid:15) = 0, (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:15) ( z ) − (cid:90) µ lk z q dzz m (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − m − z m − (cid:19) − (cid:90) µ lk z m − q dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − ( m − z m − (cid:18) − µ lk ( m − q − z m − q − + (constant) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m − | µ lk | | z | q m − q − S ϑ ∩ (∆ × (cid:36) − (0) × ∆ s ).If (cid:15) (cid:54) = 0, we can write µ lk z q dzz m − (cid:15) m = m (cid:88) j =1 a jk z − (cid:15)ζ jm dz for 0 ≤ q ≤ m − 2. Then (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:15) ( z ) − (cid:90) µ lk z q dzz m − (cid:15) m (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) j (cid:48) =1 log( z − (cid:15)ζ j (cid:48) m ) (cid:15) m − (cid:81) j (cid:48)(cid:48) (cid:54) = j (cid:48) ( ζ j (cid:48)(cid:48) m − ζ j (cid:48) m ) − (cid:90) zz m (cid:88) j =1 a jk dzz − ζ jm (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) j (cid:48) =1 log( z − (cid:15)ζ j (cid:48) m ) (cid:15) m − (cid:81) j (cid:48)(cid:48) (cid:54) = j (cid:48) ( ζ j (cid:48)(cid:48) m − ζ j (cid:48) m ) − (cid:16) m (cid:88) j =1 a jk log( z − ζ jm (cid:15) ) + (constant) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m (cid:88) j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) j (cid:48) =1 log( z − (cid:15)ζ j (cid:48) m ) (cid:81) j (cid:48)(cid:48) (cid:54) = j (cid:48) ( ζ j (cid:48)(cid:48) m − ζ j (cid:48) m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | a jk | | log( z − ζ jm (cid:15) ) || (cid:15) | m − + (constant)is bounded on each S ϑ ∩ { (cid:15) (cid:54) = 0 } .Thus τ (cid:15) ( z ) − Diag ( (cid:82) µ lk z q dzzm − (cid:15)m ) is bounded on S ϑ and the proposition follows. (cid:3) In a precise setting in the paper [15] by Hurtubise and Rousseau, they consider a linear differentialequation on P with poles along the unfolding divisor and two regular singular points ∞ H-R , R H-R . So weshould associate a relative connection ∇ (cid:48) P ×V [¯ h ] ,v l,q : ( O hol P ×V [¯ h ] ) ⊕ r −→ ( O hol P ×V [¯ h ] ) ⊕ r ⊗ Ω P ×V [¯ h ] / V [¯ h ] (cid:16) D V [¯ h ] ∪ (cid:0) {∞ H − R , R H − R } × V [¯ h ] (cid:1)(cid:17) hol such that ∇ (cid:48) P ×V [¯ h ] ,v l,q (cid:12)(cid:12) ∆ ×V [¯ h ] ∼ = ∇ P ×V [¯ h ] ,v l,q (cid:12)(cid:12) ∆ ×V [¯ h ] . In other words, we decompose the monodromy of ∇ P ×V [¯ h ] ,v l,q along ∞×V [¯ h ] to the composition of the monodromy of ∇ (cid:48) P , ¯ h,v ( l ) q around ∞ H-R and that arounda point R H-R other than ∞ H-R . The monodromy of ∇ (cid:48) around R H-R reflects the analytic continuation offundamental solutions of ∇ P ×V [¯ h ] ,v l,q along the ‘inner side’ of the unfolded divisor D V [¯ h ] . We can take afundamental solution Y (cid:48)∞ H-R ( z, ¯ h ) of ∇ (cid:48) P ×V [¯ h ] ,v l,q near ∞ H-R × V [¯ h ]. Then we can write˜ Y ϑ i ( z, ¯ h ) = Q ( z, ¯ h ) Y (cid:48)∞ H-R ( z, ¯ h ) C (cid:48)∞ H-R ,ϑ i (¯ h )for an invertible matrix Q ( z, ¯ h ) giving the isomorphism ∇ (cid:48) P ×V [¯ h ] ,v l,q | ∆ ×V [¯ h ] ∼ = ∇ P ×V [¯ h ] ,v l,q | ∆ ×V [¯ h ] . Herethe matrix C (cid:48)∞ H-R ,ϑ i (¯ h ) is a more close analogue of an unfolded Stokes matrix in [15]. Though there is anambiguity in the choice of C (cid:48)∞ H-R ,ϑ i (¯ h ) coming from the choices of ∇ (cid:48) P , V [¯ h ] ,v l,q and Y (cid:48)∞ H-R ( z, ¯ h ), we cannot say from its construction that C (cid:48)∞ H-R ,ϑ i (¯ h ) is constant in ¯ h , because we do not know the compatibility ofthe asymptotic properties between ˜ Y ϑ i ( z, ¯ h ) and ˜ Y ϑ i +1 ( z, ¯ h ) when j i (cid:54) = j i +1 .We remark that in the general setting in [14], [15], the asymptotic property of solutions of unfolded lineardifferential equations is far more complicated than our one parameter deformation case.5. Construction of an unfolded generalized isomonodromic deformation Setting of the moduli space for an unfolded generalized isomonodromic deformation. Inthis subsection, we introduce the moduli theoretic setting for describing an unfolding of the unramifiedirregular singular generalized isomonodromic deformation. Let us recall the independent variables of theusual unramified irregular singular generalized isomonodromic deformation, which basically comes from[21]. We consider unramified irregular singular connections ∇ : E −→ E ⊗ Ω C ( m t + · · · + m n t n ) and wetake a certain ´etale covering U −→ M reg g,n of the moduli stack M reg g,n of n -pointed smooth projective curvesof genus g with a universal family ( C , ˜ t , . . . , ˜ t n ) over U . Then (cid:16) Ω C /U ( m ˜ t + · · · + m n ˜ t n ) / Ω C /U (˜ t + · · · + ˜ t n ) (cid:17) r becomes the space of independent variables of the generalized isomonodromic deformation of ( E, ∇ ). Wewill give a certain perturbation of this space.First we construct a smooth covering H −→ M reg g,n of the moduli stack of n -pointed smooth projectivecurves of genus g as follows. If g = 0, we put H := Spec C , Z := P and regard Z as a curve over H . If g = 1, we put H := { D ∈ |O P (3) | | D is a smooth cubic curve } and we set Z ⊂ P × H as the universalfamily of smooth cubic curves. Assume that g ≥ 2. Then we fix l ≥ N := h ( C, ω ⊗ lC ) − C of genus g , where ω C is the canonical bundle of C . We considerthe locally closed subscheme H ⊂ Hilb P N of the Hilbert scheme which parametrizes the closed subvarieties C ⊂ P N isomorphic to the l -th canonical embeddings C (cid:44) → P ( H ( C, ω ⊗ lC )) of smooth projective curves C of genus g . Let Z ⊂ P N × H be the universal family. For any case g ≥ 0, we define a Zariski open subset H := (cid:40) ( p i ) ∈ n (cid:89) i =1 Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p i (cid:54) = p i (cid:48) for i (cid:54) = i (cid:48) (cid:41) of the fiber product (cid:81) ni =1 Z of n copies of Z over H . Similarly we define a Zariski open subset P := ( p i ) , ( p ( i ) j )) ∈ n (cid:89) i =1 Z × H n (cid:89) i =1 m i (cid:89) j =1 Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p i (cid:54) = p i (cid:48) , p i (cid:54) = p ( i (cid:48) ) j (cid:48) and p ( i ) j (cid:54) = p ( i (cid:48) ) j (cid:48) for i (cid:54) = i (cid:48) of the fiber product (cid:81) ni =1 Z × H (cid:81) ni =1 (cid:81) m i j =1 Z of n + (cid:80) ni =1 m i copies of Z over H . Then there is a canonicalprojection π P , H : P −→ H defined by π P , H (( p i ) , ( p ( i ) j )) = ( p i ) and there is a section τ H , P : H −→ P defined by τ H , P (( p i )) = (( p i ) , ( p i )).We put C := Z × H H and C P = Z × H P . Then there are universal sections σ i : P −→ C P and σ ( i ) j : P −→C P defined by σ i (( p i ) , ( p ( i ) j )) = ( p i , ( p i ) , ( p ( i ) j )), σ ( i ) j (( p i ) , ( p ( i ) j )) = ( p ( i ) j , ( p i ) , ( p ( i ) j )) which satisfy σ i ( P ) ∩ σ i (cid:48) ( P ) = ∅ , σ i ( P ) ∩ σ ( i (cid:48) ) j (cid:48) ( P ) = ∅ and σ ( i ) j ( P ) ∩ σ ( i (cid:48) ) j (cid:48) ( P ) = ∅ for i (cid:54) = i (cid:48) and any j, j (cid:48) . We define divisors D i , D ( i ) j , D ( i ) and D on C P by putting D i := σ i ( P ), D ( i ) j := σ ( i ) j ( P ), D ( i ) := (cid:80) m i j =1 D ( i ) j and D := (cid:80) ni =1 D ( i ) .We consider the closed subvariety τ H , P ( H ) ⊂ P which can be written τ H , P ( H ) = (cid:110) (( p i ) , ( p ( i ) j )) ∈ P (cid:12)(cid:12)(cid:12) p i = p ( i ) j for any i, j (cid:111) . It was necessary to set the differential form (14) in subsection 2.2 for the formulation of the moduli spaceof (˜ ν , ˜ µ )-connections. For its construction, we use the following lemma. NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 57 Lemma 5.1. Let f : X −→ S be a smooth morphism of algebraic schemes over Spec C such that all thegeometric fibers of X over S are one dimensional. Assume that X −→ S has a section σ : S −→ X .Consider the diagonals ∆ , = { ( x, y, z ) ∈ X × S X × S X | x = y } ∆ , = { ( x, y, z ) ∈ X × S X × S X | x = z } ∆ , = { ( x, y, z ) ∈ X × S X × S X | y = z } . We denote the ideal sheaf of O X × S X × S X defining ∆ i,j by I ∆ i,j . Then for each closed point p ∈ σ ( S ) ⊂ X ,there exists an affine open neighborhood W of ( p, p, p ) in X × S X × S X such that the ideal I ∆ , | W is generatedby a section z , ∈ H ( W, I ∆ , | W ) , the ideal I ∆ , | W is generated by a section z , ∈ H ( W, I ∆ , | W ) , theideal I ∆ , | W is generated by z , − z , and that z , − z , ∈ p − , ( O V ) for some open neighborhood V of ( p, p ) in X × S X .Proof. If we put s = f ( p ), the stalk of I σ ( S ) ⊗ O X s = I σ ( S ) ∩ X s at p is a principal ideal of O X s ,p . So thereis an affine open neighborhood U of p in X and a section z ∈ H ( U, I σ ( S ) | U ) such that z | U s is a generatorof I σ ( S ) ∩ U s . By Nakayama’s lemma, z becomes a generator of I σ ( S ) | U after shrinking U if necessary. Since z ⊗ ⊗ − ⊗ z ⊗ dz ⊗ ∈ I ∆ , /I , | U × S U × S U = Ω U/S ⊗ S O U is a generator after shrinking U , Nakayama’s lemma implies that z , := z ⊗ ⊗ − ⊗ z ⊗ I ∆ , | W for some affine open neighborhood W of ( p, p, p ) in X × S X × S X . If we put z , := z ⊗ ⊗ − ⊗ ⊗ z , then z , similarly becomes a generator of I ∆ , | W after shrinking W again.Since z , − z , = ( z ⊗ ⊗ − ⊗ z ⊗ − ( z ⊗ ⊗ − ⊗ ⊗ z ) = 1 ⊗ (1 ⊗ z − z ⊗ ∈ p − , ( O U × S U ) , and 1 ⊗ (1 ⊗ z − z ⊗ 1) becomes a generator of I ∆ , after shrinking W , the lemma is proved. (cid:3) Remark 5.2. In the above lemma, we may further assume that p − , ( V ) ∩ ∆ , ⊂ W and p − , ( V ) ∩ ∆ , ⊂ W .For each point h ∈ H , we consider the fiber C h of C P over τ H , P ( h ). If we put p := σ i ( τ H , P ( h )),then, by Lemma 5.1 and Remark 5.2, there is an affine open neighborhood W of p in C P and sections z ( i ) , z ( i ) j ∈ H ( W, O W ) such that z ( i ) = 0 is a defining equation of D i ∩ W , z ( i ) j = 0 is a defining equation of D ( i ) j ∩ W for each j and z ( i ) − z ( i ) j ∈ O P for any i, j . So we can take an affine open neighborhood P (cid:48) of p in P and an affine open covering {U α } of C × P P (cid:48) such that { α | D ( i ) × P P (cid:48) ⊂ U α } = { α | D i × P P (cid:48) ⊂ U α } consists of a single element α i for each i , (cid:93) { i | ( D i × P P (cid:48) ) ∩ U α (cid:54) = ∅} ≤ (cid:93) { i | ( D ( i ) × P P (cid:48) ) ∩ U α (cid:54) = ∅} ≤ α , ( D i ) P (cid:48) coincides with the zero scheme of z ( i ) ∈ H ( U α i , O U αi ), ( D ( i ) j ) P (cid:48) coincides with the zeroscheme of z ( i ) j ∈ H ( U α i , O U αi ), z ( i ) j − z ( i ) ∈ O P (cid:48) and ( z ( i ) j − z ( i ) ) | τ H , P ( H ) × P P (cid:48) = 0 for any i, j . We denotethe image of z ( i ) and z ( i ) j in O D ( i ) × P P (cid:48) by ¯ z ( i ) and ¯ z ( i ) j , respectively. We put ζ m i := exp (cid:18) π √− m i (cid:19) and consider the locus B := (cid:40) h ∈ P (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( z ( i ) j − z ( i ) ) | h = ζ jm i ( z ( i ) m i − z ( i ) ) | h for any i, j and ( z ( i ) m i − z ( i ) ) | h = ( z ( i (cid:48) ) m i (cid:48) − z ( i (cid:48) ) ) | h for any i, i (cid:48) (cid:41) which is a smooth subvariety of P (cid:48) . Note that we have z ( i ) j − z ( i ) ∈ H ( O P (cid:48) ) from the choice of P (cid:48) . If weput (cid:15) ( h ) := ( z ( i ) m i − z ( i ) )( h ) for h ∈ B , then (cid:15) : B −→ A = C is an algebraic function. There is a diagram C B (cid:47) (cid:47) (cid:32) (cid:32) B (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) A = C H and we have z ( i ) j = z ( i ) + ζ jm i (cid:15) on U α i × P B ⊂ C B .Let ( w , . . . , w s ) be a holomorphic coordinate system in a neighborhood of h in H . Then we can seethat ( z ( i ) , (cid:15), w , . . . , w s ) becomes a holomorphic coordinate system in a neighborhood of σ i ( τ H , P ( h )) in U α i × P (cid:48) B . So we can take a disk ∆ (cid:15) = { z ∈ C | | z | < (cid:15) } for small (cid:15) > 0, an analytic open neighborhood B (cid:48) of τ H , P ( h ) in (cid:15) − (∆ (cid:15) ) ⊂ B and an analytic open neighborhood U i ⊂ U α i × P (cid:48) B (cid:48) of σ i ( τ H , P ( h ))containing D ( i ) × P B (cid:48) such that U i ∩ U i (cid:48) = ∅ for i (cid:54) = i (cid:48) and(67) U i ( z ( i ) ,(cid:15),w ,...,w s ) −−−−−−−−−−→ ∼ ∆ a × ∆ (cid:15) × ∆ sr becomes biholomorphic for any i , where a, r > 0, ∆ a = { z ∈ C | | z | < a } and ∆ sr = s (cid:122) (cid:125)(cid:124) (cid:123) ∆ r × · · · × ∆ r with∆ r = { z ∈ C | | z | < r } . We define a subset Γ ( i ) j,b of the fiber C b of C × H B (cid:48) over b ∈ B (cid:48) by settingΓ ( i ) j,b := (cid:91) ≤ s ≤ (cid:110) x ∈ C b ∩ U i (cid:12)(cid:12)(cid:12) ( z ( i ) + sζ jm i (cid:15) )( x ) = 0 (cid:111) . Then Γ ( i ) j,b becomes a simple path in C b joining the two points ( D i ) b and ( D ( i ) j ) b for (cid:15) ∈ ∆ (cid:15) \ { } becauseof the bijectivity of (67). If we set Γ ( i ) j := (cid:91) b ∈B (cid:48) Γ ( i ) j,b , Γ := (cid:91) i,j Γ ( i ) j , then Γ ( i ) j and Γ are closed subsets of C × P B (cid:48) with respect to the analytic topology.We fix distinct complex numbers µ ( i )1 , . . . , µ ( i ) r ∈ C for i = 1 , . . . , n and write µ = ( µ ( i ) k ) ≤ i ≤ n ≤ k ≤ r . Then weput ϕ ( i ) µ ( T ) := ( T − µ ( i )1 )( T − µ ( i )2 ) · · · ( T − µ ( i ) r ) ∈ C [ T ] . We take an integer a ∈ Z and a tuple of complex numbers λ = ( λ ( i ) k ) ∈ C nr satisfying(i) a + n (cid:88) i =1 r (cid:88) k =1 λ ( i ) k = 0,(ii) λ ( i ) k − λ ( i ) k (cid:48) / ∈ Z for k (cid:54) = k (cid:48) .We define an algebraic variety T µ , λ over B whose set of S -valued points is given by T µ , λ ( S ) := ( ν ( i ) ( T )) ≤ i ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ( i ) ( T ) = r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) l,j ( z ( i ) ) j T l with c ( i ) l,j ∈ H ( O S )satisfying the following (a) and (b) for any noetherian scheme S over B ;(a) λ ( i ) k = r − (cid:88) l =0 c ( i ) l,m i − ( µ ( i ) k ) l for each i, k (b) ν ( i ) ( µ ( i ) k ) | p (cid:54) = ν ( i ) ( µ ( i ) k (cid:48) ) | p for k (cid:54) = k (cid:48) , 1 ≤ i ≤ n and any p ∈ D ( i ) S .Here we intend to regard ( c ( i ) l,j ) ≤ i ≤ n ≤ l ≤ r − , ≤ j ≤ m i with c ( i ) l,j ∈ H ( S, O S ) as a precise data denoted by ( ν ( i ) ( T )).We take a universal family ˜ ν ( i ) ( T ) = r − (cid:88) l =0 m − (cid:88) l =0 c ( i ) l,j ( z ( i ) ) j T l with c ( i ) l,j ∈ H ( O T µ , λ ) and write ˜ ν := (˜ ν ( i ) ( T )). If we denote by ˜ ν ( i ) s , ( c ( i ) l,j ) s the restrictions of ˜ ν ( i ) , c ( i ) l,j to s ∈ T µ , λ , respectively, we can see by Lemma 2.1 that (cid:88) p ∈D ( i ) s res p (cid:18) ˜ ν ( i ) s ( µ ( i ) k ) d ¯ z ( i ) (¯ z ( i ) ) m i − (cid:15) m i (cid:19) = r − (cid:88) l =0 m i − (cid:88) j =0 ( c ( i ) l,j ) s ( µ ( i ) k ) l (cid:88) p ∈D ( i ) s res z ( i ) = p (cid:18) ( z ( i ) ) j dz ( i ) ( z ( i ) ) m i − (cid:15) m i (cid:19) = − r − (cid:88) l =0 m i − (cid:88) j =0 ( c ( i ) l,j ) s ( µ ( i ) k ) l res z ( i ) = ∞ (cid:18) ( z ( i ) ) j dz ( i ) ( z ( i ) ) m i − (cid:15) m i (cid:19) = r − (cid:88) l =0 ( c ( i ) l,m i − ) s ( µ ( i ) k ) l . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 59 So the equality (a) in the definition of T µ , λ means the equality(68) λ ( i ) k = (cid:88) p ∈D ( i ) s res p (cid:18) ˜ ν ( i ) s ( µ ( i ) k ) d ¯ z ( i ) (¯ z ( i ) ) m i − (cid:15) m i (cid:19) where (cid:80) p ∈D ( i ) s runs over the set theoretical points p of D ( i ) s . For each point p = (cid:15)ζ jm ∈ D ( i ) s , we have˜ ν ( i ) s ( µ ( i ) k ) (cid:12)(cid:12) p = r − (cid:88) l =0 m i − (cid:88) j (cid:48) =0 ( c ( i ) l,j (cid:48) ) s ( (cid:15) ( s ) ζ jm ) j (cid:48) ( µ ( i ) k ) l for 1 ≤ k ≤ r . The condition (b) in the definition of T µ , λ is that r − (cid:88) l =0 m i − (cid:88) j (cid:48) =0 ( c ( i ) l,j (cid:48) ) s ( (cid:15) ( s ) ζ jm ) j (cid:48) ( µ ( i ) k ) l (cid:54) = r − (cid:88) l =0 m i − (cid:88) j (cid:48) =0 ( c ( i ) l,j (cid:48) ) s ( (cid:15) ( s ) ζ jm ) j (cid:48) ( µ ( i ) k (cid:48) ) l for k (cid:54) = k (cid:48) , when (cid:15) ( s ) (cid:54) = 0 and that r − (cid:88) l =0 ( c ( i ) l, ) s ( µ k ) l (cid:54) = r − (cid:88) l =0 ( c ( i ) l, ) s ( µ k (cid:48) ) l for k (cid:54) = k (cid:48) when (cid:15) ( s ) = 0.By Theorem 2.11, there is a relative moduli space(69) π T µ , λ : M α C , D (˜ ν , µ ) −→ T µ , λ of (˜ ν , µ )-connections over T µ , λ . Note that the morphism π T µ , λ in (69) is an algebraic smooth morphism ofquasi-projective schemes. We consider the pull-back diagram M α C , D (˜ ν , µ ) × B B (cid:48) −−−−→ M α C , D (˜ ν , µ ) (cid:121) (cid:121) B (cid:48) −−−−→ B where the horizontal arrows are open immersions as analytic spaces.5.2. Unramified irregular singular generalized isomonodromic deformation. The unramified ir-regular singular generalized isomonodromic deformation is the well-known theory by Jimbo, Miwa andUeno, which is completely given in [21], [22], [23] with explicit calculations using formal solutions basedon the Malgrange-Sibuya theorem ([2, Theorem 4.5.1]). We recall here a moduli theoretic construction ofthe unramified irregular singular generalized isomonodromic deformation given in [19], which is valid in ahigher genus case.Recall that there are compositions of morphisms M α C , D (˜ ν , µ ) −→ T µ , λ −→ B (cid:15) −→ ∆ (cid:15) . We consider thefibers B (cid:15) =0 := B × ∆ (cid:15) { } , T µ , λ ,(cid:15) =0 := T µ , λ × B B (cid:15) =0 , M α C , D (˜ ν , µ ) (cid:15) =0 := M α C , D (˜ ν , µ ) × B B (cid:15) =0 over (cid:15) = 0 ∈ ∆ (cid:15) . Then π T µ , λ ,(cid:15) =0 : M α C , D (˜ ν , µ ) (cid:15) =0 −→ T µ , λ ,(cid:15) =0 is the relative moduli space of unramifiedirregular singular connections. In our moduli theoretic setting, the unramified irregular singular generalizedisomonodromic deformation is given in [19, Theorem 6.2] as an algebraic splittingΨ : π ∗T µ , λ ,(cid:15) =0 T T µ , λ ,(cid:15) =0 −→ T M α C , D (˜ ν , µ ) (cid:15) =0 of the canonical surjection T M α C , D (˜ ν , µ ) (cid:15) =0 dπ T µ , λ ,(cid:15) =0 −−−−−−−→ ( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 . Here we use the symbol Ψ insteadof the symbol D used in [19], for the purpose of avoiding confusion with the divisor of singularity of theconnection.Let us recall the construction of Ψ . For each Zariski open subset T (cid:48) ⊂ T µ , λ ,(cid:15) =0 and for each vector field v ∈ H ( T (cid:48) , T T µ , λ ,(cid:15) =0 | T (cid:48) ), let T (cid:48) [ v ] := T (cid:48) × Spec C [ h ] / ( h ) I v −→ T (cid:48) be the corresponding morphism satisfying I v ⊗ C [ h ] / ( h ) = id T (cid:48) . If we put ν ( i )0 ,hor ( T ) := r − (cid:88) l =0 m i − (cid:88) j =0 ( I ∗ v ( c ( i ) l,j ) T (cid:48) − ¯ hv (( c ( i ) l.j ) T (cid:48) ))( z ( i ) ) j T l ν ( i )0 ,v ( T ) := r − (cid:88) l =0 m i − (cid:88) j =0 v (( c ( i ) l,j ) T (cid:48) )( z ( i ) ) j T l , then we have I ∗ v (˜ ν ( i ) ( T )) = ν ( i )0 ,hor ( T ) + ¯ hν ( i )0 ,v ( T ) and ν ( i )0 ,hor ( T ) is the pullback of ˜ ν ( i ) ( T ) via the trivialprojection T (cid:48) [ v ] = T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) (cid:44) → T µ , λ . We consider the fiber product C T (cid:48) [ v ] = C T × T (cid:48) T (cid:48) [ v ]with respect to I v : T (cid:48) [ v ] −→ T (cid:48) and the trivial projection C T (cid:48) −→ T (cid:48) . We denote the pullback of z ( i ) underthe morphism C T (cid:48) [ v ] = C T (cid:48) × T (cid:48) ( T (cid:48) × Spec C [ h ] / ( h )) −→ C T (cid:48) by ˜ z ( i ) .For some ´etale surjective morphism ˜ M −→ M α C , D (˜ ν , µ ), there is a universal family ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) on C ˜ M . We put ˜ M (cid:48) := ˜ M × T µ , λ T (cid:48) , ˜ M (cid:48) [ v ] := ˜ M × T µ , λ T (cid:48) [ v ] and denote the restriction of ( ˜ E, ˜ ∇ , { ˜ N ( i ) } )to C ˜ M (cid:48) by ( ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } ). In the following definition, C ˜ M (cid:48) [ v ] means the fiber product C T (cid:48) × T (cid:48) ˜ M (cid:48) [ v ]with respect to the canonical morphism C T (cid:48) −→ T (cid:48) and the composition ˜ M (cid:48) [ u ] −→ T (cid:48) [ u ] I v −→ T (cid:48) . On theother hand, relative differentials in Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) are with respect to the composition C ˜ M (cid:48) [ v ] −→ ˜ M (cid:48) [ v ] =˜ M (cid:48) × Spec C [ h ] / ( h ) −→ ˜ M (cid:48) of the trivial projections. Definition 5.3. (cid:0) E v , ∇ v , {N ( i )0 ,v } (cid:1) is a horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respect to v if(1) E v is an algebraic vector bundle on C ˜ M (cid:48) [ v ] of rank r ,(2) ∇ v : E v −→ E v ⊗ Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) ( D ˜ M (cid:48) [ v ] ) is a morphism of sheaves satisfying ∇ v ( f a ) = a ⊗ df + f ∇ v ( a )for f ∈ O hol C ˜ M (cid:48) v ] and a ∈ E v ,(3) ∇ v is integrable in the sense that the restriction of ∇ v to any open set U [ v ] ⊂ C ˜ M (cid:48) [ v ] \ D ˜ M (cid:48) [ v ] satisfying E v | U [ v ] ∼ = (cid:0) O U [ v ] (cid:1) ⊕ r is expressed by (cid:0) O U [ v ] (cid:1) ⊕ r (cid:51) f ...f r (cid:55)→ df ...df r + (cid:16) ˜ Ad ˜ z + Bdh (cid:17) f ...f r ∈ (cid:0) O U [ v ] (cid:1) ⊕ r ⊗ Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) ( D ˜ M (cid:48) [ v ] )satisfying d (cid:16) ˜ Ad ˜ z + Bdh (cid:17) + (cid:104)(cid:16) ˜ Ad ˜ z + Bdh (cid:17) , (cid:16) ˜ Ad ˜ z + Bdh (cid:17)(cid:105) = 0 in Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) (2 D ˜ M (cid:48) [ v ] ),(4) N ( i )0 ,v : E v | D ( i )˜ M (cid:48) v ] −→ E v | D ( i )˜ M (cid:48) v ] is an endomorphism satisfying ϕ ( i ) µ ( N ( i )0 ,v ) = 0,(5) the relative connection ∇ v defined by the composition ∇ v : E v ∇ v −−→ E v ⊗ Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) ( D ˜ M (cid:48) [ v ] ) −→ E v ⊗ Ω C ˜ M (cid:48) v ] / ˜ M (cid:48) [ v ] ( D ˜ M (cid:48) [ v ] )satisfies ( ν ( i )0 ,hor + ¯ hν ( i )0 ,v )( N ( i )0 ,v ) d ˜ z ( i ) (˜ z ( i ) ) m i = ∇ v (cid:12)(cid:12) D ( i )˜ M (cid:48) v ] for any i and(6) (cid:0) E v , ∇ v , {N ( i )0 ,v } (cid:1) ⊗ O ˜ M (cid:48) [ v ] / ¯ h O ˜ M (cid:48) [ v ] ∼ = (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) .The following proposition is essentially given in the proof of [19, Theorem 6.2] and we omit its proofhere. Proposition 5.4. There exists a unique horizontal lift (cid:0) E v , ∇ v , {N ( i )0 ,v } (cid:1) of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respectto v For each vector field v ∈ H ( T (cid:48) , T T µ , λ ,(cid:15) =0 | T (cid:48) ), the horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respect to v induces a relative connection (cid:0) E v , ∇ v , {N ( i )0 ,v } (cid:1) which gives a morphism ˜ M (cid:48) [ v ] −→ M α C , D (˜ ν , µ ) making the NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 61 diagram ˜ M (cid:48) [ v ] −→ M α C , D (˜ ν , µ ) (cid:121) (cid:121) T (cid:48) [ v ] I v −→ T (cid:48) (cid:44) → T µ , λ commutative. This morphism corresponds to a section of T M α C , D (˜ ν , µ ) (cid:15) =0 ⊗ O ˜ M (cid:48) over ˜ M (cid:48) which descends toa vector field Φ ( v ) ∈ H (cid:0) π − T µ , λ ,(cid:15) =0 ( T (cid:48) ) , T M α C , D (˜ ν , µ ) (cid:15) =0 (cid:12)(cid:12) π − T µ , λ ,(cid:15) =0 ( T (cid:48) ) (cid:1) . We can show that the correspondence T T µ , λ ,(cid:15) =0 (cid:51) v (cid:55)→ Φ ( v ) ∈ ( π T µ , λ ,(cid:15) =0 ) ∗ T M α C , D (˜ ν , µ ) (cid:15) =0 is an O T µ , λ ,(cid:15) =0 -homomorphism. We omit its proof because it is the same as that of Proposition 5.14 whichis given later. So Φ is equivalent to the morphism(70) Ψ : ( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 −→ T M α C , D (˜ ν , µ ) (cid:15) =0 . We devote the rest of this subsection to the proof of the integrability of the subbundle im Ψ ⊂ T M α C , D (˜ ν , µ ) (cid:15) =0 . The integrability of the irregular singular generalized isomonodromic deformation in thezero genus case is proved by Jimbo, Miwa and Ueno in [21, Theorem 4.2], which is extended by Bremerand Sage in [8, Theorem 5.1]. Although the integrability is almost a consequence of the Malgrange-Sibuyaisomorphism [2, Theorem 4.5.1] in a general case as in [7], it will be worth giving a proof of the integrabilityof Ψ , because the situation in an unfolded case is different.For the proof of the integrability condition of Ψ , we extend the definition of horizontal lift given inDefinition 5.3. We consider a morphism u : T (cid:48) [ u ] := T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) ⊂ T µ , λ ,(cid:15) =0 satisfying u ⊗ C [ h , h ] / ( h , h ) = id T (cid:48) and write u ∗ ˜ ν ( i ) ( T ) = ν ( i ) hor ( T ) + ν ( i )1 ( T )¯ h + ν ( i )2 ( T )¯ h + ν ( i )1 , ( T )¯ h ¯ h where ν ( i ) hor ( T ) is the pullback of ˜ ν ( i ) ( T ) by the composition T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) (cid:44) → T µ , λ ofthe trivial projection and the inclusion and ν ( i )1 ( T ) , ν ( i )2 ( T ) , ν ( i )1 , ( T ) are pullbacks of polynomials in O D ( i ) T (cid:48) [ T ]via the trivial projection T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) .We consider the fiber product ˜ M (cid:48) [ u ] := ˜ M (cid:48) × T (cid:48) T (cid:48) [ u ] with respect to the canonical morphism ˜ M (cid:48) −→ T (cid:48) and T (cid:48) [ u ] u −→ T (cid:48) . We can extend the notion of horizontal lift given in Definition 5.3 to the morphism u : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) .We say that a tuple (cid:0) E u , ∇ u , {N ( i )0 ,u } (cid:1) is a horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respect to u if E u is a locally free sheaf on C ˜ M (cid:48) [ u ] , ∇ u : E u −→ E u ⊗ Ω C ˜ M (cid:48) u ] / ˜ M (cid:48) ( D ˜ M (cid:48) [ u ] ) is an integrable connection and N ( i )0 ,u : E u (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] −→ E u (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] is an endomorphism such that the conditions (3), (4), (5) and (6) of Definition5.3 hold after replacing v by u . Then we have the following: Lemma 5.5. There exists a unique horizontal lift (cid:0) E u , ∇ u , {N ( i )0 ,u } (cid:1) of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respect to u .Proof. We consider the restriction of ˜ ∇ ˜ M (cid:48) to an affine open neighborhood U ( i ) of D ( i ) (cid:15) =0 such that ˜ E ˜ M (cid:48) (cid:12)(cid:12) U ( i ) ∼ = O ⊕ rU ( i ) . It can be written˜ ∇ ˜ M (cid:48) (cid:12)(cid:12) U ( i ) : O ⊕ rU ( i ) (cid:51) f ...f r (cid:55)→ df ...df r + A ( z ( i ) ) dz ( i ) ( z ( i ) ) m i f ...f r ∈ O ⊕ rU ( i ) ⊗ Ω U ( i ) / ˜ M (cid:48) ( D ( i )˜ M (cid:48) ) . Here we may assume that A ( z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) = Diag (˜ ν ( µ k )) (cid:12)(cid:12) D ( i )˜ M (cid:48) . We can take a lift A (˜ z ( i ) ) of A ( z ( i ) ) as a matrix of algebraic functions on U ( i )˜ M (cid:48) [ u ] satisfying(71) ∂A (˜ z ( i ) ) ∂ ¯ h = ∂A (˜ z ( i ) ) ∂ ¯ h = 0 . Indeed for an arbitrary lift ˜ A (˜ z ( i ) ) of A ( z ( i ) ), we can write d ˜ A = A d ˜ z ( i ) + A d ¯ h + A d ¯ h with respect tothe identification Ω U ( i )˜ M (cid:48) u ] / ˜ M (cid:48) = O U ( i )˜ M (cid:48) u ] d ˜ z ( i ) ⊕ O U ( i )˜ M (cid:48) u ] d ¯ h ⊕ O U ( i )˜ M (cid:48) u ] d ¯ h . Here relative differential forms in Ω U ( i )˜ M (cid:48) u ] / ˜ M (cid:48) are with respect to the composition of the trivial projections U ( i )˜ M (cid:48) [ u ] −→ ˜ M (cid:48) [ u ] = ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) . Then the replacement A (˜ z ( i ) ) := ˜ A − ¯ h A − ¯ h A + ¯ h ¯ h ∂A ∂ ¯ h satisfies the condition (71) because of the equalities ∂A ∂ ¯ h = ∂ ˜ A∂ ¯ h ∂ ¯ h = ∂A ∂ ¯ h . We put B (˜ z ( i ) ) := Diag ( (cid:82) ν ( i )1 ( µ k ) d ˜ z ( i )(˜ z ( i )) mi ) , B (˜ z ( i ) ) := Diag ( (cid:82) ν ( i )2 ( µ k ) d ˜ z ( i )(˜ z ( i )) mi ) ,B , (˜ z ( i ) ) := Diag ( (cid:82) ν ( i )1 , ( µ k ) d ˜ z ( i )(˜ z ( i )) mi ) . Note that (˜ z ( i ) ) m i − B (˜ z ( i ) ), (˜ z ( i ) ) m i − B (˜ z ( i ) ) and (˜ z ( i ) ) m i − B , (˜ z ( i ) ) are matrices of polynomials in ˜ z ( i ) ,because ν ( i )1 ( µ k ) d ˜ z ( i ) (˜ z ( i ) ) m i , ν ( i )2 ( µ k ) d ˜ z ( i ) (˜ z ( i ) ) m i and ν ( i )1 , ( µ k ) d ˜ z ( i ) (˜ z ( i ) ) m i have no residue part. If we define C (˜ z ( i ) ) d ˜ z ( i ) (˜ z ( i ) ) m i := dB (˜ z ( i ) ) + (cid:2) A (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i C (˜ z ( i ) ) d ˜ z ( i ) (˜ z ( i ) ) m i := dB (˜ z ( i ) ) + (cid:2) A (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i , we have C (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = Diag ( ν ( i )1 ( µ k )) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] and C (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = Diag ( ν ( i )2 ( µ k )) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] . Since B (˜ z ( i ) ), B (˜ z ( i ) ), dB (˜ z ( i ) ) and dB (˜ z ( i ) ) commute to each other, we have (cid:2) C (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i = (cid:20) dB (˜ z ( i ) ) + (cid:2) A (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i , B (˜ z ( i ) ) (cid:21) = (cid:2)(cid:2) A (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i = (cid:2)(cid:2) A (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i = (cid:2) C (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i . If we put C (˜ z ( i ) ) := (cid:2) C (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) = (cid:2) C (˜ z ( i ) ) , B (˜ z ( i ) ) (cid:3) , then we can see that C (˜ z ( i ) ) is a matrix of algebraic functions on U ( i )˜ M (cid:48) [ u ] such that C (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = 0. Wecan check the integrability of η = ( A + ¯ h C + ¯ h C + ¯ h ¯ h C ) d ˜ z ( i ) (˜ z ( i ) ) m i + B d ¯ h + B d ¯ h by the calculation dη + [ η, η ] = ( C + ¯ h C ) d ¯ h ∧ d ˜ z ( i ) (˜ z ( i ) ) m i + ( C + ¯ h C ) d ¯ h ∧ d ˜ z ( i ) (˜ z ( i ) ) m i + dB ∧ d ¯ h + dB ∧ d ¯ h + (cid:0) [ A, B ] + ¯ h [ C , B ] (cid:1) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h + (cid:0) [ A, B ] + ¯ h [ C , B ] (cid:1) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h = (cid:18) dB + ( − C + [ A, B ]) d ˜ z ( i ) (˜ z ( i ) ) m i (cid:19) ∧ d ¯ h + (cid:18) dB + ( − C + [ A, B ]) d ˜ z ( i ) (˜ z ( i ) ) m i (cid:19) ∧ d ¯ h + ¯ h ( − C + [ C , B ]) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h + ¯ h ( − C + [ C , B ]) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h = 0 . If we put C , (˜ z ( i ) ) d ˜ z ( i ) (˜ z ( i ) ) m i := dB , (˜ z ( i ) ) + (cid:2) A (˜ z ( i ) ) , B , (˜ z ( i ) ) (cid:3) d ˜ z ( i ) (˜ z ( i ) ) m i , then the connection matrix˜ η := η + ¯ h ¯ h C , d ˜ z ( i ) (˜ z ( i ) ) m i + ¯ h B , (˜ z ( i ) ) d ¯ h + ¯ h B , (˜ z ( i ) ) d ¯ h satisfies the integrability condition d ˜ η + [˜ η, ˜ η ] = dη + [ η, η ] + ¯ h C , d ¯ h ∧ d ˜ z ( i ) (˜ z ( i ) ) m i + ¯ h C , d ¯ h ∧ d ˜ z ( i ) (˜ z ( i ) ) m i + ¯ h dB , ∧ d ¯ h + ¯ h dB , ∧ d ¯ h + ¯ h [ A, B , ] d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h + ¯ h [ A, B , ] d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h = ¯ h (cid:18) dB , + ( − C , + [ A, B , ]) d ˜ z ( i ) (˜ z ( i ) ) m i (cid:19) ∧ d ¯ h + ¯ h (cid:18) dB , + ( − C , + [ A, B , ]) d ˜ z ( i ) (˜ z ( i ) ) m i (cid:19) ∧ d ¯ h = 0 . Then the connection ∇ uU ( i ) : O ⊕ rU ( i )˜ M (cid:48) u ] −→ O ⊕ rU ( i )˜ M (cid:48) u ] ⊗ Ω U ( i )˜ M (cid:48) u ] / ˜ M (cid:48) ( D ( i )˜ M (cid:48) [ u ] )given by the connection matrix˜ η = (cid:16) A (˜ z ( i ) ) + ¯ h C (˜ z ( i ) ) + ¯ h C (˜ z ( i ) ) + ¯ h ¯ h (cid:0) C (˜ z ( i ) ) + C , (˜ z ( i ) ) (cid:1)(cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i + (cid:0) B (˜ z ( i ) ) + ¯ h B , (˜ z ( i ) ) (cid:1) d ¯ h + (cid:0) B (˜ z ( i ) ) + ¯ h B , (˜ z ( i ) ) (cid:1) d ¯ h becomes an integrable connection. If we put N ( i ) U ( i ) ,u := Diag ( µ k ) , then (cid:16) O ⊕ rU ( i )˜ M (cid:48) u ] , ∇ uU ( i ) , N ( i ) U ( i ) ,u (cid:17) is a localhorizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1)(cid:12)(cid:12) U ( i ) .Assume that (cid:16) O ⊕ rU ( i )˜ M (cid:48) u ] , ∇ (cid:48) , N (cid:48) (cid:17) is another local horizontal lift given by a connection matrix( A (˜ z ( i ) ) + ¯ h C (cid:48) (˜ z ( i ) ) + ¯ h C (cid:48) (˜ z ( i ) ) + ¯ h ¯ h C (cid:48) , (˜ z ( i ) )) d ˜ z ˜ z m i + B (cid:48) (˜ z ( i ) ) d ¯ h + B (cid:48) (˜ z ( i ) ) d ¯ h + B (cid:48) , (˜ z )¯ h d ¯ h + B (cid:48) , (˜ z ( i ) )¯ h d ¯ h . We want to construct an isomorphism between ∇ uU ( i ) and ∇ (cid:48) . Since C (cid:48) (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) [ u ] , C (cid:48) (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) [ u ] and C (cid:48) , (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) [ u ] are diagonal matrices by the assumption, the integrability condition − (cid:0) ( C (cid:48) + ¯ h C (cid:48) , ) d ¯ h + ( C (cid:48) + ¯ h C (cid:48) , ) d ¯ h (cid:1) ∧ d ˜ z ( i ) (˜ z ( i ) ) m i = (cid:0) dB (cid:48) ( z ( i ) ) + ¯ h dB (cid:48) , ( z ( i ) ) (cid:1) ∧ d ¯ h + (cid:0) dB (cid:48) ( z ( i ) ) + ¯ h dB (cid:48) , ( z ( i ) ) (cid:1) ∧ d ¯ h + (cid:16) B (cid:48) , ( z ( i ) ) − B (cid:48) , ( z ( i ) ) + [ B (cid:48) (˜ z ( i ) ) , B (cid:48) (˜ z ( i ) )] (cid:17) d ¯ h ∧ d ¯ h + (cid:16)(cid:2) A (˜ z ( i ) ) , B (cid:48) (˜ z ( i ) ) + ¯ h B (cid:48) , ( z ( i ) ) (cid:3) + ¯ h (cid:2) C (cid:48) , B (cid:48) (˜ z ( i ) ) (cid:3)(cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h + (cid:16)(cid:2) A (˜ z ( i ) ) , B (cid:48) (˜ z ( i ) ) + ¯ h B (cid:48) , ( z ( i ) ) (cid:3) + ¯ h (cid:2) C (cid:48) , B (cid:48) (˜ z ( i ) ) (cid:3)(cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i ∧ d ¯ h implies dB (cid:48) (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = Diag (cid:0) ν ( i )1 ( µ k ) d ˜ z ( i )(˜ z ( i )) mi (cid:1)(cid:12)(cid:12)(cid:12) D ( i )˜ M (cid:48) u ] and dB (cid:48) (˜ z ( i ) ) (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = Diag (cid:0) ν ( i )2 ( µ k ) d ˜ z ( i )(˜ z ( i )) mi (cid:1)(cid:12)(cid:12)(cid:12) D ( i )˜ M (cid:48) u ] .Then B (˜ z ( i ) ) − B (cid:48) (˜ z ( i ) ), B (˜ z ( i ) ) − B (cid:48) (˜ z ( i ) ) are matrices of algebraic functions on U ( i ) [ u ] and applyingthe transform ( I r + ¯ h ( B (˜ z ( i ) ) − B (cid:48) (˜ z ( i ) )) + ¯ h ( B (˜ z ( i ) ) − B (cid:48) (˜ z ( i ) )) to ∇ (cid:48) , we may assume that B (cid:48) = B , B (cid:48) = B and consequently, C (cid:48) = dB + [ A, B ] d ˜ z ( i ) (˜ z ( i ) ) m i = C and C (cid:48) = dB + [ A, B ] d ˜ z ( i ) (˜ z ( i ) ) m i = C .Since [ B , B ] = 0, we have B (cid:48) , = B (cid:48) , and C (cid:48) , = dB (cid:48) , + ([ A, B (cid:48) , ] + [ C , B ]) d ˜ z ( i ) (˜ z ( i ) ) m i implies that dB (cid:48) , (cid:12)(cid:12) D ( i )˜ M (cid:48) u ] = Diag (cid:0) ν ( i )1 , ( µ k ) d ˜ z ( i )(˜ z ( i )) mi (cid:1) . So we can see that B , − B (cid:48) , is a matrix of regular functions on U ( i ) [ u ] and the transform I r + ¯ h ¯ h ( B , − B (cid:48) , ) gives an isomorphism between (cid:16) O ⊕ rU ( i )˜ M (cid:48) u ] , ∇ uU ( i ) , N ( i ) U ( i ) ,u (cid:17) and (cid:16) O ⊕ rU ( i )˜ M (cid:48) u ] , ∇ (cid:48) , N (cid:48) (cid:17) . We can see that such an isomorphism is unique because it is determined by thecoefficients of d ¯ h and d ¯ h .If an affine open subset U ⊂ C ˜ M (cid:48) is disjoint from D ˜ M (cid:48) , then we can easily give a local horizontal liftof (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1)(cid:12)(cid:12) U . In that case { ˜ N ( i )˜ M (cid:48) } (cid:12)(cid:12) U is nothing. Patching local horizontal lifts altogether, weobtain a unique horizontal lift (cid:0) E u , ∇ u , {N ( i )0 ,u } (cid:1) of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { ˜ N ( i )˜ M (cid:48) } (cid:1) with respect to u . (cid:3) Theorem 5.6. The subbundle Ψ (( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 ) ⊂ T M α C , D (˜ ν , µ ) (cid:15) =0 determined by (70) satisfies theintegrability condition (cid:2) Ψ (( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 ) , Ψ (( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 ) (cid:3) ⊂ Ψ (( π T µ , λ ,(cid:15) =0 ) ∗ T T µ , λ ,(cid:15) =0 ) . Proof. Take a Zariski open set T (cid:48) ⊂ T µ , λ ,(cid:15) =0 and vector fields v , v ∈ H ( T (cid:48) , T T (cid:48) ). We will prove theequality(72) [Φ ( v ) , Φ ( v )] = Φ ([ v , v ])from which the theorem follows immediately. Let T (cid:48) × C [ h , h ] / ( h , h ) ˜ I v −−→ T (cid:48) × Spec C [ h , h ] / ( h , h )be the morphism over Spec C [ h , h ] / ( h , h ) corresponding to the ring homomorphism˜ I ∗ v : O T (cid:48) [ h , h ] / ( h , h ) (cid:51) f + f ¯ h + f ¯ h + f , ¯ h ¯ h (cid:55)→ f + ( f + v ( f ))¯ h + f ¯ h + ( f , + v ( f ))¯ h ¯ h ∈ O T (cid:48) [ h , h ] / ( h , h )and let T (cid:48) × C [ h , h ] / ( h , h ) ˜ I v −−→ T (cid:48) × Spec C [ h , h ] / ( h , h ) be the morphism corresponding to the ringhomomorphism˜ I ∗ v : O T (cid:48) [ h , h ] / ( h , h ) (cid:51) f + f ¯ h + f ¯ h + f , ¯ h ¯ h (cid:55)→ f + f ¯ h + ( f + v ( f ))¯ h + ( f , + v ( f ))¯ h ¯ h ∈ O T (cid:48) [ h , h ] / ( h , h ) . NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 65 By the calculation f + f ¯ h + f ¯ h + f , ¯ h ¯ h I ∗ v (cid:55)→ f + f ¯ h + ( f + v ( f ))¯ h + ( f , + v ( f ))¯ h ¯ h I ∗ v (cid:55)→ f + ( f + v ( f ))¯ h + ( f + v ( f ))¯ h + ( f , + v ( f ) + v ( f ) + v v ( f ))¯ h ¯ h I ∗− v (cid:55)→ f + ( f + v ( f ))¯ h + f ¯ h + ( f , + v ( f ) + v v ( f ) − v v ( f ))¯ h ¯ h I ∗− v (cid:55)→ f + f ¯ h + f ¯ h + ( f , + ( v v − v v )( f ))¯ h ¯ h , we can see that the composition ˜ I ∗− v ˜ I ∗− v ˜ I ∗ v ˜ I ∗ v is given by˜ I ∗− v ˜ I ∗− v ˜ I ∗ v ˜ I ∗ v : O T (cid:48) [ h , h ] / ( h , h ) (cid:51) f + f ¯ h + f ¯ h + f , ¯ h ¯ h (cid:55)→ f + f ¯ h + f ¯ h + ( f , + ( v v − v v )( f ))¯ h ¯ h ∈ O T (cid:48) [ h , h ] / ( h , h ) . (73)The vector field Φ ( v ) corresponds to a morphism ˜ M (cid:48) × Spec C [ h ] / ( h ) −→ ˜ M (cid:48) . This morphismtogether with the second projection ˜ M (cid:48) × Spec C [ h ] / ( h ) −→ Spec C [ h ] / ( h ) gives a morphism(74) ˜ M (cid:48) × Spec C [ h ] / ( h ) −→ ˜ M (cid:48) × Spec C [ h ] / ( h )over Spec C [ h ] / ( h ). Let(75) ˜ I Φ ( v ) : ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) . be the base change of (74) under the projection Spec C [ h , h ] / ( h , h ) −→ Spec C [ h ] / ( h ). Similarly wecan define a morphism(76) ˜ I Φ ( v ) : ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) × Spec C [ h , h ] / ( h , h )from the morphism ˜ M (cid:48) × Spec C [ h ] / ( h ) −→ ˜ M (cid:48) corresponding to Φ ( v ). We can see by a similarcalculation to that of (73) that the composition ˜ I ∗ Φ ( − v ) ˜ I ∗ Φ ( − v ) ˜ I ∗ Φ ( v ) ˜ I ∗ Φ ( v ) corresponds to the ring ho-momorphism˜ I ∗ Φ ( − v ) ˜ I ∗ Φ ( − v ) ˜ I ∗ Φ ( v ) ˜ I ∗ Φ ( v ) : O ˜ M (cid:48) [ h , h ] / ( h , h ) (cid:51) f + f ¯ h + f ¯ h + f , ¯ h ¯ h (77) (cid:55)→ f + f ¯ h + f ¯ h + ( f , + (Φ ( v )Φ ( v ) − Φ ( v )Φ ( v ))( f ))¯ h ¯ h ∈ O ˜ M (cid:48) [ h , h ] / ( h , h ) . Let π T (cid:48) : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) be the first projection. By Lemma 5.5, there exists a uniquehorizontal lift (cid:16) E π T (cid:48) ◦ ˜ I v , ∇ π T (cid:48) ◦ ˜ I v , (cid:8) N ( i )0 ,π T (cid:48) ◦ ˜ I v (cid:9)(cid:17) of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { N ( i )˜ M (cid:48) } (cid:1) with respect to the composition π T (cid:48) ◦ ˜ I v : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) . Then we can see that(id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( v ) ) ∗ (cid:16) E π T (cid:48) ◦ ˜ I v , ∇ π T (cid:48) ◦ ˜ I v , (cid:8) N ( i )0 ,π T (cid:48) ◦ ˜ I v (cid:9)(cid:17) is a horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { N ( i )˜ M (cid:48) } (cid:1) , in the sense of Lemma 5.5, with respect to the composition π T (cid:48) ◦ ˜ I v ◦ ˜ I v ◦ ˜ I − v ◦ ˜ I − v : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) . Let ρ : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) × Spec C [ h ] / ( h )be the morphism whose corresponding ring homomorphism ρ ∗ : O T (cid:48) [ h ] / ( h ) −→ O T (cid:48) [ h , h ] / ( h , h ) isgiven by ρ ∗ ( f + gh ) = f + g ¯ h ¯ h for f, g ∈ O T (cid:48) . Then we have π T (cid:48) ◦ ˜ I v ◦ ˜ I v ◦ ˜ I − v ◦ ˜ I − v = I [ v ,v ] ◦ ρ, where I [ v ,v ] : T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) means the morphism corresponding to the commutator vector field[ v , v ] = v v − v v . If we denote by (cid:0) E [ v ,v ]0 , ∇ [ v ,v ]0 , {N ( i )0 , [ v ,v ] } (cid:1) the horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { N ( i )˜ M (cid:48) } (cid:1) ,in the sense of Proposition 5.4, with respect to the the commutator vector field [ v , v ] ∈ H ( T (cid:48) , T T (cid:48) ), we cansee that (id × ρ ) ∗ (cid:0) E [ v ,v ]0 , ∇ [ v ,v ]0 , {N ( i )0 , [ v ,v ] } (cid:1) is also a horizontal lift of (cid:0) ˜ E ˜ M (cid:48) , ˜ ∇ ˜ M (cid:48) , { N ( i )˜ M (cid:48) } (cid:1) , in the sense of Lemma 5.5, with respect to I [ v ,v ] ◦ ρ = π T (cid:48) ◦ ˜ I v ◦ ˜ I v ◦ ˜ I − v ◦ ˜ I − v : T (cid:48) × Spec C [ h , h ] / ( h , h ) −→ T (cid:48) .By the uniqueness of the horizontal lift in Lemma 5.5, we have an isomorphism(id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( v ) ) ∗ (cid:16) E π T (cid:48) ◦ ˜ I v , ∇ π T (cid:48) ◦ ˜ I v , {N ( i )0 ,π T (cid:48) ◦ ˜ I v } (cid:17) (78) ∼ = (id × ρ ) ∗ (cid:0) E [ v ,v ]0 , ∇ [ v ,v ]0 , {N ( i )0 , [ v ,v ] } (cid:1) . The flat family (id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( − v ) ) ∗ (id × ˜ I Φ ( v ) ) ∗ (cid:16) E π T (cid:48) ◦ ˜ I v , ∇ π T (cid:48) ◦ ˜ I v , {N ( i )0 ,π T (cid:48) ◦ ˜ I v } (cid:17) associatedto (78) corresponds to the composition π ˜ M (cid:48) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( − v ) ◦ ˜ I Φ ( − v ) : ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) , where π ˜ M (cid:48) : ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) is the first projection. The same associated flat family(id × ρ ) ∗ (cid:0) E [ v ,v ]0 , ∇ [ v ,v ]0 , {N ( i )0 , [ v ,v ] } (cid:1) induced by (78) corresponds to the composition π ˜ M (cid:48) ◦ ˜ I Φ ([ v ,v ]) ◦ (id × ρ ) : ˜ M (cid:48) × Spec C [ h , h ] / ( h , h ) −→ ˜ M (cid:48) . Thus we have π ˜ M (cid:48) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( − v ) ◦ ˜ I Φ ( − v ) = π ˜ M (cid:48) ◦ ˜ I Φ ([ v ,v ]) ◦ (id × ρ ). We can see by (77)that the morphism π ˜ M (cid:48) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( v ) ◦ ˜ I Φ ( − v ) ◦ ˜ I Φ ( − v ) is given by the ring homomorphism O ˜ M (cid:48) (cid:51) f (cid:55)→ f + (Φ ( v )Φ ( v ) − Φ ( v )Φ ( v ))¯ h ¯ h ∈ O ˜ M (cid:48) [ h , h ] / ( h , h ) . On the other hand, the morphism π ˜ M (cid:48) ◦ ˜ I Φ ([ v ,v ]) ◦ (id × ρ ) is given by the ring homomorphism O ˜ M (cid:48) (cid:51) f (cid:55)→ f + Φ ( v v − v v )¯ h ¯ h ∈ O ˜ M (cid:48) [ h , h ] / ( h , h ) . Hence we have Φ ( v )Φ ( v ) − Φ ( v )Φ ( v ) = Φ ( v v − v v ), which is nothing but the equation (72) andthe proposition is proved. (cid:3) Global horizontal lift in the unfolded case and the proof of Theorem 0.1. In this subsection,we give an analytic local lift of the unramified irregular singular generalized isomonodromic deformationgiven in subsection 5.2. The key point is to construct a global horizontal lift via patching local horizontallifts given in Proposition 4.11. The consequent global horizontal lift given in Proposition 5.11 produces theproof of Theorem 0.1.Take a point x ∈ M α C , D (˜ ν , µ ) (cid:15) =0 × B B (cid:48) which corresponds to a ( ν , µ )-connection ( E, ∇ , { N ( i ) } ). Recallthat we are given an analytic open subset U i ⊂ C B (cid:48) with a biholomorphic map(79) U i ∼ −→ ∆ a × ∆ (cid:15) × ∆ sr given by (67) in subsection 5.1. We take a loop ˜ γ x in ( U i ) x ⊂ C x which is a boundary of a disk containing D ( i ) x . We consider the morphism ∇ † : E nd ( E ) (cid:51) u (cid:55)→ ∇ ◦ u − u ◦ ∇ ∈ E nd ( E ) ⊗ Ω C x ( D x )and assume the following: Assumption 5.7. (1) The monodromy of ∇ : E −→ E ⊗ Ω C x ( D x ) along ˜ γ x has the r distinct eigen-values and(2) H (( U i ) x , ker ∇ † | ( U i ) x ) = C .There is an ´etale morphism ˜ M −→ M α C , D (˜ ν , µ ) whose image contains x such that there is a univer-sal family ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) on ( C , D ) ˜ M over ˜ M . We can take an analytic open neighborhood M ◦ of x in M α C , D (˜ ν , µ ) × B B (cid:48) with a factorization M ◦ −→ ˜ M −→ M α C , D (˜ ν , µ ). We denote by (cid:0) ˜ E holM ◦ , ˜ ∇ holM ◦ , { ˜ N ( i ) ,holM ◦ } (cid:1) the pullback of ( ˜ E, ˜ ∇ , { ˜ N ( i ) } ) to ( C , D ) M ◦ .In the following, we successively replace M ◦ by its shrink till Definition 5.8. After shrinking M ◦ , wemay assume that the morphism induced by (79)( U i ) M ◦ ∼ −→ ∆ a × M ◦ NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 67 is an isomorphism. We denote the image of M ◦ under the morphism M α C , D (˜ ν , µ ) × B B (cid:48) −→ T µ , λ × B B (cid:48) by T ◦ , which is an analytic open subset of T µ , λ × B B (cid:48) . Then the inclusion T ◦ (cid:44) → T µ , λ × B B (cid:48) (cid:44) → T µ , λ corresponds to a tuple of polynomials ν = ( ν ( i ) ( T )) ≤ i ≤ n given by ν ( i ) ( T ) = r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) l,j ( z ( i ) ) j T l with c ( i ) l,j ∈ H ( T ◦ , O hol T ◦ ) satisfying (a) and (b) of the definition of T µ , λ .We apply the process in subsection 4.2 to the restricted relative connection (cid:0) ˜ E holM ◦ , ˜ ∇ holM ◦ , { ˜ N ( i ) ,holM ◦ } (cid:1)(cid:12)(cid:12) ( U i ) M ◦ .Using Proposition 4.3, there is an isomorphism θ ( i ) : ˜ E holM ◦ (cid:12)(cid:12) ( U i ) M ◦ ∼ −→ ( O hol ( U i ) M ◦ ) ⊕ r after shrinking M ◦ suchthat the connection ( θ ( i ) ⊗ id) ◦ ˜ ∇ holM ◦ (cid:12)(cid:12) ( U i ) M ◦ ◦ ( θ ( i ) ) − is canonically extended to a global relative connection ∇ ( i ) , P M ◦ : ( O hol P × M ◦ ) ⊕ r −→ ( O hol P × M ◦ ) ⊕ r ⊗ Ω P × M ◦ /M ◦ ( D M ◦ ∪ ( {∞} × M ◦ )) hol , where we are assuming the identification ( U i ) M ◦ = ∆ a × M ◦ (cid:44) → P × M ◦ . Let A ( i ) ( z ( i ) , (cid:15) ) dz ( i ) ( z ( i ) ) m i − (cid:15) m i = m i − (cid:88) j =0 A ( i ) j ( (cid:15) )( z ( i ) ) j dz ( i ) ( z ( i ) ) m i − (cid:15) m i be the connection matrix of ∇ ( i ) , P M ◦ . By Assumption 5.7, we can see, after shrinking M ◦ , that m i − (cid:92) j =0 ker (cid:16) ad( A ( i ) j ( (cid:15) )) (cid:17) = O holM ◦ in the same way as (58) in subsection 4.2. As in the argument in subsection 4.2 producing (49), we cantake matrices Ξ ( i ) l,j ( z ( i ) ) of polynomials in z ( i ) of degree less than m i satisfying(80) A ( i ) ( z ( i ) , (cid:15) ) = r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) l,j Ξ ( i ) l,j ( z ( i ) )and ( z ( i ) ) j θ ( i ) ◦ (cid:0) ˜ N ( i ) ,hol (cid:1) l ◦ ( θ ( i ) ) − (cid:12)(cid:12) D ( i ) M ◦ = Ξ ( i ) l,j ( z ( i ) ) (cid:12)(cid:12) D ( i ) M ◦ . Indeed there is a polynomial(81) ψ ( i ) ( T ) = a ( i ) r − ( z ( i ) ) T r − + · · · + a ( i )1 ( z ( i ) ) T + a ( i )0 ( z ( i ) ) ∈ O hol V [ z ( i ) ][ T ]in T of degree less than r with each a ( i ) k ( z ( i ) ) ∈ O V [ z ( i ) ] a polynomial in z ( i ) of degree less than m i andΞ ( i ) l,j ( z ( i ) ) is obtained from ( z ( i ) ) j ψ ( i ) ( A ( z ( i ) , (cid:15) )) l by substituting (cid:15) m i in ( z ( i ) ) m i .By Lemma 4.9, we can take an adjusting data (cid:0) R ( i ) , ( l ) j,l (cid:48) (cid:1) for the connection ∇ ( i ) , P M ◦ after shrinking M ◦ . Ifwe put ˜Ξ ( i ) l,j ( z ( i ) ) := Ξ ( i ) l,j ( z ( i ) ) − m i − (cid:88) q =0 (cid:88) ≤ l (cid:48) ≤ m i − − q (cid:2) A ( i ) q ( (cid:15) ) , R ( i ) , ( l ) j,l (cid:48) (cid:3) ( z ( i ) ) q + l (cid:48) (82) − m i − (cid:88) q =0 (cid:88) m i − q ≤ l (cid:48) ≤ m i − (cid:2) A ( i ) q ( (cid:15) ) , R ( i ) , ( l ) j,l (cid:48) (cid:3) (cid:15) m i ( z ( i ) ) q + l (cid:48) − m i , then we have res z ( i ) = ∞ (cid:18) ˜Ξ ( i ) l,j ( z ( i ) ) dz ( i ) ( z ( i ) ) m i − (cid:15) m i (cid:19) = 0and ˜Ξ ( i ) l,j ( z ( i ) ) (cid:12)(cid:12) D ( i ) M ◦ = Ξ ( i ) l,j ( z ( i ) ) (cid:12)(cid:12) D ( i ) M ◦ − (cid:104) A ( i ) ( z ( i ) ) , m i − (cid:88) l (cid:48) =0 R ( i ) , ( l ) j,l (cid:48) ( z ( i ) ) l (cid:48) (cid:105)(cid:12)(cid:12)(cid:12) D ( i ) M ◦ . We consider the relative connection(83) ∇ P × M ◦ [¯ h ] ,v ( i ) l,j : ( O hol P × M ◦ [¯ h ] ) ⊕ r −→ ( O hol P × M ◦ [¯ h ] ) ⊕ r ⊗ Ω P × M ◦ [¯ h ] /M ◦ [¯ h ] (cid:16) D ( i ) M ◦ [¯ h ] ∪ ( ∞ × M ◦ [¯ h ]) (cid:17) hol determined by the connection matrix (cid:16) A ( z ( i ) , (cid:15) ) + ¯ h ˜Ξ ( i ) l,j ( z ( i ) ) (cid:17) dz ( i ) ( z ( i ) ) m i − (cid:15) m i , where we write M ◦ [¯ h ] := M ◦ × Spec C [ h ] / ( h ). By Proposition 4.11, we can take a horizontal lift(84) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j : ( O hol P × M ◦ [¯ h ] ) ⊕ r −→ ( O hol P × M ◦ [¯ h ] ) ⊕ r ⊗ ( ι M ◦ [¯ h ] ) ∗ Ω P × M ◦ \ Γ M ◦ )[¯ h ] (cid:14) M ◦ (cid:0) ∞ × M ◦ [¯ h ] (cid:1) hol of ∇ P × M ◦ [¯ h ] ,v ( i ) l,j given by a connection matrix (cid:0) A ( z ( i ) , (cid:15) ) + ¯ h ˜Ξ ( i ) l,j ( z ( i ) ) (cid:1) dz ( i ) ( z ( i ) ) m i − (cid:15) m i + B ( i ) l,j ( z ( i ) ) d ¯ h, where ι M ◦ [¯ h ] : ( P × M ◦ \ Γ M ◦ )[¯ h ] (cid:44) → P × M ◦ [¯ h ] is the canonical inclusion. By the construction, therestriction of ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j to P × M ◦ [¯ h ] × ∆ (cid:15) Spec C [ (cid:15) ] / ( (cid:15) m i ) coincides with the horizontal lift giving theunramified irregular singular generalized isomonodromic deformation. Definition 5.8. We call the collection (cid:16) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:17) ≤ i ≤ n ≤ l ≤ r − , ≤ j ≤ m i − of integrable connections deter-mined by (cid:0) ˜Ξ ( i ) l,j ( z ( i ) ) (cid:1) , (cid:0) B ( i ) l,j ( z ( i ) ) (cid:1) in (84) a block of local horizontal lifts. Take an analytic open subset T (cid:48) ⊂ T ◦ ⊂ T µ , λ × B B (cid:48) and a ∆ (cid:15) -relative holomorphic vector field v ∈ H ( T (cid:48) , T hol T (cid:48) / ∆ (cid:15) ) on T (cid:48) . Then v corresponds to an analytic morphism I v : T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) (cid:44) → T µ , λ × B B (cid:48) over ∆ (cid:15) satisfying I v | T (cid:48) × Spec C [ h ] / ( h ) = id T (cid:48) . We put T (cid:48) [ v ] := T (cid:48) × Spec C [ h ] / ( h ) which is regarded as ananalytic space over T (cid:48) via I v and consider the fiber product C T (cid:48) [ v ] := C T (cid:48) × T (cid:48) ( T (cid:48) × Spec C [ h ] / ( h )) −−−−→ C T (cid:48) := C × P T (cid:48) (cid:121) (cid:121) T (cid:48) × Spec C [ h ] / ( h ) I v −−−−→ T (cid:48) of C T (cid:48) −→ T (cid:48) and T (cid:48) × Spec C [ h ] / ( h ) I v −→ T (cid:48) . The morphism I v corresponds to an analytic morphism I v B(cid:48) : T (cid:48) × Spec C [ h ] / ( h ) −→ B (cid:48) over ∆ (cid:15) and a tuple of polynomials(85) ν hor + ¯ h ν v = ( ν ( i ) hor ( T ) + ¯ hν ( i ) v ( T ))where ν ( i ) hor ( T ) ∈ O hol D ( i ) T (cid:48) [ h ] / ( h ) [ T ] and ν ( i ) v ( T ) ∈ O hol D ( i ) T (cid:48) [ T ] are given by ν ( i ) hor ( T ) = r − (cid:88) l =0 m i − (cid:88) j =0 ( I ∗ v c ( i ) l,j − ¯ hv ( c ( i ) l,j ))(˜ z ( i ) ) j T l ν ( i ) v ( T ) = r − (cid:88) l =0 m i − (cid:88) j =0 v ( c ( i ) l,j )(˜ z ( i ) ) j T l . Here ˜ z ( i ) j is the pull-back of z ( i ) j under the morphism C T (cid:48) [ v ] id × I v −−−−→ C T (cid:48) −→ C B (cid:48) and ν ( i ) hor ( T ) + ¯ hν ( i ) v ( T ) ∈O hol D ( i ) T (cid:48) [ h ] / ( h ) [ T ] should satisfy (a) in the definition of T µ , λ in subsection 5.1. For an analytic open subset U ⊂ C T (cid:48) , we denote by U [ v ] the open subspace of C T (cid:48) [ v ] whose underlying set of points is U .We consider the sheaf of T (cid:48) -relative differential forms (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ v ]/ T (cid:48) (cid:1) hol with respect to the compositeof the trivial projections C T (cid:48) [ v ] = C × P T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 69 which is different from the structure of C T (cid:48) [ v ] over T (cid:48) coming from the fiber product structure. Note that (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ v ]/ T (cid:48) (cid:1) hol is locally generated by d ˜ z and d ¯ h , where ˜ z is the pullback of a uniformizing parameter z of C T (cid:48) via the first projection C T (cid:48) × T (cid:48) T (cid:48) [ v ] −→ C T (cid:48) . Let ι ( C T (cid:48) \ Γ T (cid:48) )[ v ] : ( C T (cid:48) \ Γ T (cid:48) )[ v ] (cid:44) → C T (cid:48) [ v ] be the inclusion morphism. We denote by ι C T (cid:48) \ Γ T (cid:48) : C T (cid:48) \ Γ T (cid:48) (cid:44) → C T (cid:48) its restriction to the underlying setsof points. Definition 5.9. We define the O hol C T (cid:48) [ v ] -subsheaf Ω C T (cid:48) ,v of ( ι ( C T (cid:48) \ Γ T (cid:48) )[ v ] ) ∗ (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ v ]/ T (cid:48) (cid:1) hol by the condi-tion that Ω C T (cid:48) ,v is locally generated by d ˜ z ( i ) (˜ z ( i ) ) m i − (cid:15) m i and (cid:0) ι C T (cid:48) \ Γ T (cid:48) (cid:1) ∗ (cid:0) O hol C T (cid:48) \ Γ T (cid:48) (cid:1) d ¯ h around points in Γ ( i ) T (cid:48) [ v ] and locally generated by d ˜ z and d ¯ h around points in ( C T (cid:48) \ Γ T (cid:48) )[ v ] where z is a local holomorphic coordinate of C T (cid:48) \ Γ T (cid:48) . We denote by Ω C T (cid:48) ,v the canonical image of Ω C T (cid:48) ,v ∧ Ω C T (cid:48) ,v in ( ι ( C T (cid:48) \ Γ T (cid:48) )[ v ] ) ∗ (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ v ]/ T (cid:48) (cid:1) hol . We put M (cid:48) := M ◦ × T ◦ T (cid:48) and consider the analytic space M (cid:48) [ v ] := M (cid:48) × Spec C [ h ] / ( h ) with thestructure morphisms M (cid:48) [ v ] := M (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) × Spec C [ h ] / ( h ) I v −→ T (cid:48) . We denote the base change of C × P T (cid:48) , D × P T (cid:48) and D ( i ) × P T (cid:48) via M (cid:48) [ v ] −→ T (cid:48) by C M (cid:48) [ v ] , D M (cid:48) [ v ] and D ( i ) M (cid:48) [ v ] , respectively. We denote the pullback of a local holomorphic coordinate z of C T (cid:48) under the morphism C M (cid:48) [ v ] −→ C T (cid:48) by ˜ z .Let us consider the analytic open subspace ( U i ) M (cid:48) [ v ] ⊂ C M (cid:48) [ v ] = C T (cid:48) × T (cid:48) ( M (cid:48) × Spec C [ h ] / ( h )). Using(67) in subsection 5.1, we have an analytic isomorphism( U i ) M (cid:48) [ v ] ∼ = ∆ a × M (cid:48) [ v ] = ∆ a × M (cid:48) × Spec C [ h ] / ( h )whose structure morphism over T µ , λ is given by∆ a × M (cid:48) × Spec C [ h ] / ( h ) −→ M (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) × Spec C [ h ] / ( h ) I v −→ T (cid:48) (cid:44) → T µ , λ . We remark that the elements in Ω C T (cid:48) ,v ⊗ O T (cid:48) [ v ] O M (cid:48) [ v ] ⊂ ( ι ( C M (cid:48) \ Γ M (cid:48) )[ v ] ) ∗ (cid:0) Ω C M (cid:48) \ Γ M (cid:48) )[ v ]/ M (cid:48) (cid:1) hol are relativedifferentials with respect to the morphism C M (cid:48) [ v ] = C T (cid:48) × T (cid:48) ( M (cid:48) × Spec C [ h ] / ( h )) −→ M (cid:48) × Spec C [ h ] / ( h ) −→ M (cid:48) , where the arrows are the trivial projections. The restriction of the above morphism to ( U i ) M (cid:48) [ v ] is justthe trivial projection ( U i ) M (cid:48) [ v ] ∼ = ∆ a × M (cid:48) × Spec C [ h ] / ( h ) −→ M (cid:48) . The corresponding inclusion O holM (cid:48) (cid:44) →O hol ( U i ) M (cid:48) [ v ] induces the ring homomorphism O holM (cid:48) [˜ z ( i ) ] −→ O hol ( U i ) M (cid:48) [ v ] from the polynomial ring. We denote the image of a matrix A ( z ( i ) ) of polynomials with coefficients in O holM (cid:48) under this ring homomorphism by A (˜ z ( i ) ).We denote the restriction of (cid:0) ˜ E holM ◦ , ˜ ∇ holM ◦ , { ˜ N ( i ) ,holM ◦ } (cid:1) to C M (cid:48) by (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) . Definition 5.10. We say that a tuple (cid:0) E v , ∇ v , {N ( i ) v } (cid:1) is a horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) withrespect to v ∈ H ( T (cid:48) , T hol T (cid:48) / ∆ (cid:15) ) and with respect to blocks of local horizontal lifts (cid:0) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:1) if(1) E v is a rank r holomorphic vector bundle on C M (cid:48) [ v ] ,(2) ∇ v : E v −→ E v ⊗ O hol CT (cid:48) [ v ] Ω C T (cid:48) ,v is a morphism of sheaves satisfying ∇ v ( f a ) = a ⊗ df + f ∇ v ( a ) for f ∈ O hol C M (cid:48) [ v ] and a ∈ E v ,(3) ∇ v is integrable in the sense that the restriction of ∇ v to any open set U [ v ] ⊂ ( C M (cid:48) \ Γ M (cid:48) )[ v ] whichis expressed by E v | U [ v ] ∼ = (cid:16) O holU [ v ] (cid:17) ⊕ r (cid:51) f ...f r (cid:55)→ df ...df r + (cid:0) A d ˜ z + B dh (cid:1) f ...f r ∈ (cid:16) O holU [ v ] (cid:17) ⊕ r ⊗ O CT (cid:48) [ v ] Ω C T (cid:48) ,v satisfies d (cid:0) A d ˜ z + B dh (cid:1) + (cid:0) A d ˜ z + B dh (cid:1) ∧ (cid:0) A d ˜ z + B dh (cid:1) = 0 in E nd (cid:0) ( O holU [ v ] ) ⊕ r (cid:1) ⊗ O CT (cid:48) [ v ] Ω C T (cid:48) ,v ,(4) N ( i ) v : E v | D ( i ) M (cid:48) [ v ] −→ E v | D ( i ) M (cid:48) [ v ] is an endomorphism satisfying ϕ ( i ) µ ( N ( i ) v ) = 0,(5) the relative connection ∇ v defined by the composition ∇ v : E v ∇ v −−→ E v ⊗ O hol CT (cid:48) [ v ] Ω C T (cid:48) ,v −→ E v ⊗ Ω C M (cid:48) [ v ] /M (cid:48) [ v ] ( D M (cid:48) [ v ] ) hol satisfies ( ν ( i ) hor + ¯ hν ( i ) v )( N ( i ) v ) d ˜ z ( i ) (˜ z ( i ) ) m i − (cid:15) m i = ∇ v (cid:12)(cid:12) D ( i ) M (cid:48) [ v ] for any i ,(6) (cid:0) E v , ∇ v , {N ( i ) v } (cid:1) ⊗ O holM (cid:48) [ v ] /h O holM (cid:48) [ v ] ∼ = (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) ,(7) there is an isomorphism θ ( i ) ,v : E v (cid:12)(cid:12) ( U i ) M (cid:48) [ v ] ∼ −→ ( O hol ( U i ) M (cid:48) [ v ] ) ⊕ r which is a lift of the restriction θ ( i ) | ( U i ) M (cid:48) of the given isomorphism θ ( i ) : ˜ E | ( U i ) M ◦ ∼ −→ ( O hol ( U i ) M ◦ ) ⊕ r such that the consequent con-nection matrix of ( θ ( i ) ,v ⊗ id) ◦ ∇ v ◦ ( θ ( i ) ,v ) − is given by (cid:16) A ( i ) ( z ( i ) , (cid:15) ) + ¯ h r − (cid:88) l =0 m i − (cid:88) j =0 v ( c ( i ) l,j ) ˜Ξ ( i ) l,j ( z ( i ) ) (cid:17) dz ( i ) ( z ( i ) ) m i − (cid:15) m i + r − (cid:88) l =0 m i − (cid:88) j =0 v ( c ( i ) l,j ) B ( i ) l,j ( z ( i ) ) d ¯ h. The following proposition on the existence of a global horizontal lift is a key process in the constructionof an unfolded generalized isomonodromic deformation. Proposition 5.11. For any ∆ (cid:15) -relative holomorphic vector field v ∈ H ( T (cid:48) , T hol T ◦ / ∆ (cid:15) ) , there exists aunique horizontal lift (cid:0) E v , ∇ v , { N ( i ) v } (cid:1) of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to v and with respect to theblocks of local horizontal lifts (cid:16) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:17) .Proof. We can take an analytic open covering { U β } of C M (cid:48) which is a refinement of {U α × P (cid:48) M (cid:48) } such that U β is contractible and ˜ E holM (cid:48) (cid:12)(cid:12) U β ∼ = ( O holU β ) ⊕ r for any β . Moreover, we may assume that U β ∩ Γ ( i ) M (cid:48) = ∅ unless U β = ( U i ) M (cid:48) . Recall that ( θ ( i ) ⊗ id) ◦ ˜ ∇ holM (cid:48) (cid:12)(cid:12) ( U i ) M (cid:48) ◦ ( θ ( i ) ) − is canonically extended to a global connection ∇ ( i ) , P M ◦ (cid:12)(cid:12) P × M (cid:48) : ( O hol P × M (cid:48) ) ⊕ r −→ ( O hol P × M (cid:48) ) ⊕ r ⊗ Ω P × M (cid:48) /M (cid:48) ( D M (cid:48) ∪ ( {∞} × M (cid:48) )) hol given by the connection matrix A ( i ) ( z ( i ) , (cid:15) ) dz ( i ) ( z ( i ) ) m i − (cid:15) m i . Here we use the identification ( U i ) M (cid:48) = ∆ a × M (cid:48) (cid:44) → P × M (cid:48) . As in Definition 5.8, there is a block (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) of local horizontal lifts given by (cid:0) ˜Ξ ( i ) l,j ( z ( i ) ) (cid:1) and (cid:0) B ( i ) l,j ( z ( i ) ) (cid:1) . We put A ( i ) v ( z ( i ) ) := r − (cid:88) l =0 m i − (cid:88) j =0 v ( c ( i ) l,j ) ˜Ξ ( i ) l,j ( z ( i ) ) B ( i ) v ( z ( i ) ) := r − (cid:88) l =0 m i − (cid:88) j =0 v ( c ( i ) l,j ) B ( i ) l,j ( z ( i ) )and denote by ι M (cid:48) [¯ h ] : (cid:0) P × M (cid:48) \ Γ M (cid:48) (cid:1) [¯ h ] (cid:44) → P × M (cid:48) [¯ h ] the inclusion morphism. Consider the connection(86) ∇ flat P × M (cid:48) [¯ h ] ,v : ( O hol P × M (cid:48) [¯ h ] ) ⊕ r −→ ( O hol P × M (cid:48) [¯ h ] ) ⊕ r ⊗ ( ι M (cid:48) [¯ h ] ) ∗ Ω P × M (cid:48) \ Γ M (cid:48) )[¯ h ] (cid:14) M (cid:48) ( ∞ × M (cid:48) ) hol determined by the connection matrix (cid:16) A ( i ) (˜ z ( i ) , (cid:15) ) + ¯ hA v ( z ( i ) ) (cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i − (cid:15) m i + B ( i ) v ( z ( i ) ) d ¯ h. Then we can see by the same calculation as in the proof of Proposition 4.11 that ∇ flat P × M (cid:48) [¯ h ] ,v is an inte-grable connection. We denote by N ( i ) v the substitution of (cid:15) m i for ( z ( i ) ) m i in ψ ( i ) (cid:0) A ( i ) (˜ z ( i ) , (cid:15) ) + ¯ hA v ( z ( i ) ) (cid:1) , NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 71 where ψ ( i ) is given in (81). Then (cid:0) O ⊕ r ( U i ) M (cid:48) [ v ] , ∇ flat P × M (cid:48) [¯ h ] ,v (cid:12)(cid:12) ( U i ) M (cid:48) [ v ] , (cid:8) N ( i ) v (cid:9)(cid:1) gives a local horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1)(cid:12)(cid:12) ( U i ) M (cid:48) with respect to v .Assume that U β ∩ D ( i ) M (cid:48) = ∅ for any i . Then the connection ˜ ∇ holM (cid:48) (cid:12)(cid:12) U β is given by a connection matrix A ( z ) dz , for some local holomorphic coordinate z of C T (cid:48) over T (cid:48) . We can take a matrix ˜ A (˜ z ) with entries in O holU β [ v ] which is a lift of A ( z ), where ˜ z is the pullback of z under the morphism C T (cid:48) [ v ] id × I v −−−−→ C T (cid:48) . We canwrite d ˜ A (˜ z ) = C (˜ z ) d ˜ z + B ( z ) d ¯ h. If we put ˜ A (cid:48) (˜ z ) := ˜ A (˜ z ) − ¯ hB ( z ), then we have d ˜ A (cid:48) (˜ z ) ∈ M r ( O holU β [ v ] ) d ˜ z and ∇ vβ : ( O holU β [ v ] ) ⊕ r −→ ( O holU β [ v ] ) ⊕ r ⊗ Ω C T (cid:48) ,v f ...f r (cid:55)→ df ...df r + ˜ A (cid:48) (˜ z ) d ˜ z f ...f r becomes a flat connection. So (( O holU β [ v ] ) ⊕ r , ∇ vβ ) becomes a local horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1)(cid:12)(cid:12) U β ,where { ˜ N ( i ) ,holM (cid:48) } (cid:12)(cid:12) U β is nothing in this case.From the above arguments, we obtain a local horizontal lift ( E vβ , ∇ vβ , {N vβ } ) of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1)(cid:12)(cid:12) U β for each piece U β of the covering C M (cid:48) = (cid:83) β U β . If U β (cid:54) = U β (cid:48) , then Γ M (cid:48) ∩ U β ∩ U β (cid:48) = ∅ by the assumption.Assume that ∇ vβ is given by( O holU β [ v ] ) ⊕ r ∼ −→ E vβ ∇ vβ −−→ E vβ ⊗ Ω C T (cid:48) ,v ∼ −→ ( O holU β [ v ] ) ⊕ r ⊗ Ω C T (cid:48) ,v f ...f r (cid:55)→ df ...df r + ( ˜ A β (˜ z ) d ˜ z + B β ( z ) d ¯ h ) f ...f r , where the integrability condition − ∂ ˜ A β (˜ z ) ∂ ¯ h d ˜ z ∧ d ¯ h + dB β ( z ) ∧ d ¯ h + ( ˜ A β (˜ z ) B β ( z ) − B β ( z ) ˜ A β (˜ z )) d ˜ z ∧ d ¯ h = 0is satisfied and so for ∇ uβ (cid:48) . There is an invertible matrix P β,β (cid:48) ( z ) of holomorphic functions on U ββ (cid:48) = U β ∩ U β (cid:48) satisfying P β,β (cid:48) ( z ) − dP β,β (cid:48) ( z ) + P β,β (cid:48) ( z ) − ˜ A β ( z ) dzP β,β (cid:48) ( z ) = ˜ A β (cid:48) ( z ) dz coming from the isomorphism ( E vβ , ∇ vβ ) (cid:12)(cid:12) U ββ (cid:48) ∼ −→ ( ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) ) (cid:12)(cid:12) U ββ (cid:48) ∼ −→ ( E vβ (cid:48) , ∇ vβ (cid:48) ) (cid:12)(cid:12) U ββ (cid:48) . We can take a matrix˜ P ββ (cid:48) (˜ z, ¯ h ) of holomorphic functions on U ββ (cid:48) [¯ h ] which is a lift of P ββ (cid:48) ( z ). If we put˜ A (cid:48) β (˜ z ) d ˜ z + B (cid:48) β ( z ) d ¯ h := ˜ P β,β (cid:48) (˜ z, ¯ h ) − d ˜ P β,β (cid:48) (˜ z, ¯ h ) + ˜ P β,β (cid:48) (˜ z, ¯ h ) − (cid:0) ˜ A β (˜ z ) d ˜ z + B β ( z ) d ¯ h (cid:1) ˜ P β,β (cid:48) (˜ z, ¯ h ) , then we can write ˜ A β (cid:48) (˜ z ) = ˜ A (cid:48) β (˜ z ) + ¯ hC β ( z ). If we put Q ββ (cid:48) ( z ) := B β (cid:48) ( z ) − B (cid:48) β ( z ), then Q ββ (cid:48) ( z ) isholomorphic on U β ∩ U β (cid:48) = ( U β ∩ U β (cid:48) ) \ (Γ M (cid:48) ∩ U β ∩ U β (cid:48) ) and we have( I r + ¯ hQ ββ (cid:48) ( z )) − d ( I r + ¯ hQ ββ (cid:48) ( z )) + ( I r + ¯ hQ ββ (cid:48) ( z )) − ( ˜ A (cid:48) β (˜ z ) d ˜ z + B (cid:48) β ( z ) d ¯ h )( I r + ¯ hQ ββ (cid:48) ( z ))= ¯ hdQ ββ (cid:48) + Q ββ (cid:48) d ¯ h + ˜ A β (cid:48) (˜ z ) d ˜ z − ¯ hC β ( z ) d ˜ z + ¯ h [ ˜ A β (cid:48) (˜ z ) , B β (cid:48) ( z ) − B (cid:48) β ( z )] d ˜ z + B (cid:48) β ( z ) d ¯ h = ˜ A β (cid:48) (˜ z ) d ˜ z − ¯ hC β ( z ) d ˜ z + ¯ h (cid:0) dB β (cid:48) ( z ) + [ ˜ A β (cid:48) (˜ z ) , B β (cid:48) ( z )] d ˜ z (cid:1) − ¯ h (cid:0) dB (cid:48) β ( z ) + [ ˜ A β (cid:48) (˜ z ) , B (cid:48) β ( z )] d ˜ z (cid:1) + (cid:0) Q ββ (cid:48) ( z ) + B (cid:48) β ( z ) (cid:1) d ¯ h = ˜ A β (cid:48) (˜ z ) d ˜ z − ¯ hC β ( z ) d ˜ z + ¯ h (cid:32) ∂ ˜ A β (cid:48) ∂ ¯ h (˜ z ) − ∂ ˜ A (cid:48) β (˜ z ) ∂ ¯ h (˜ z ) (cid:33) d ˜ z + B β (cid:48) ( z ) d ¯ h = ˜ A β (cid:48) (˜ z ) d ˜ z + B β (cid:48) ( z ) d ¯ h Thus the composition of P β,β (cid:48) (˜ z, ¯ h ) with I r + ¯ hQ ββ (cid:48) ( z ) gives an isomorphism between ( E vβ , ∇ vβ ) (cid:12)(cid:12) U ββ (cid:48) [ v ] and( E vβ (cid:48) , ∇ vβ (cid:48) ) (cid:12)(cid:12) U ββ (cid:48) [ v ] whose restriction to U ββ (cid:48) = U ββ (cid:48) [ v ] ⊗ C [¯ h ] / (¯ h ) is the identity. By construction, we cansee that this isomorphism is unique, because it is essentially determined by the d ¯ h -coefficients. So we can patch ( E vβ , ∇ vβ , {N vβ } ) together and obtain a global horizontal lift ( E v , ∇ v , { N ( i ) v } ) of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to v and with respect to the blocks (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) of local horizontal lifts. Since the localhorizontal lift is unique up to a unique isomorphism, we can see that a global horizontal lift ( E v , ∇ v , { N ( i ) v } )is unique up to an isomorphism. (cid:3) For a vector field v ∈ H ( T (cid:48) , T hol T µ , λ × B B (cid:48) / ∆ (cid:15) ) over an analytic open subset T (cid:48) ⊂ T ◦ ⊂ T µ , λ × B B (cid:48) , wehave by Proposition 5.11 a unique horizontal lift ( E v , ∇ v , {N ( i ) v } ) of the restriction (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) of the universal family to C × H M (cid:48) with respect to v and with respect to the blocks (cid:0) ∇ flat P × M ◦ [¯ h ] ,v ( i ) l,j (cid:1) of localhorizontal lifts. Let ∇ v : E v ∇ v −−→ E v ⊗ Ω C T (cid:48) ,v −→ E v ⊗ Ω C M (cid:48) [ v ] /M (cid:48) [ v ] ( D M (cid:48) [ v ] )be the relative connection induced by ∇ v . Then ( E v , ∇ v , {N ( i ) v } ) becomes a holomorphic flat family of( ν , µ )-connections on C M (cid:48) [ v ] over M (cid:48) [ v ], which determines a morphism M (cid:48) [ v ] −→ M α C , D (˜ ν , µ ) × T µ , λ T (cid:48) making the diagram(87) M (cid:48) [ v ] −−−−→ M α C , D (˜ ν , µ ) × T µ , λ T (cid:48) (cid:121) (cid:121) T (cid:48) [ v ] I v −−−−→ T (cid:48) commutative. This morphism corresponds to a vector field Φ( v ) ∈ H (cid:0) ( π ◦ ) − ( T (cid:48) ) , T holM ◦ / ∆ (cid:15) (cid:12)(cid:12) ( π ◦ ) − ( T (cid:48) ) (cid:1) ,where π ◦ : M ◦ −→ T ◦ is the projection morphism. We can see dπ ◦ (Φ( v )) = v by the commutative diagram(87), where dπ ◦ : π ◦∗ T holM ◦ / ∆ (cid:15) −→ T hol T ◦ / ∆ (cid:15) is the differential of π ◦ . Thus we have defined a map(88) Φ : T hol T ◦ / ∆ (cid:15) (cid:51) v (cid:55)→ Φ( v ) ∈ ( π ◦ ) ∗ T holM ◦ / ∆ (cid:15) . In the rest of this subsection, we will prove that the correspondence (88) defined above is an O hol T ◦ -homomorphism. In order to prove it, we extend the notion of horizontal lift.Let C [ I ] = C ⊕ I be a finite dimensional local algebra over C with the maximal ideal I satisfying I = 0. For a morphism u : T (cid:48) × Spec C [ I ] −→ T (cid:48) over ∆ (cid:15) satisfying u | T (cid:48) × Spec C [ I ] /I = id T (cid:48) , we write T (cid:48) [ u ] := T (cid:48) × Spec C [ I ] which is endowed with the structure morphism u : T (cid:48) [ u ] −→ T (cid:48) . We endow thefiber product C T (cid:48) [ u ] := C × H T (cid:48) × Spec C [ I ] with the structure morphism C T (cid:48) [ u ] = C × H T (cid:48) × Spec C [ I ] −→ T (cid:48) × Spec C [ I ] u −→ T (cid:48) . For an analytic open subset U ⊂ C T (cid:48) , we denote by U [ u ] the open subspace of C T (cid:48) [ u ] whose underlying setof points is U .We consider the sheaf of differential forms (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ u ]/ T (cid:48) (cid:1) hol with respect to the composite of thetrivial projections C T (cid:48) [ u ] = C × P T (cid:48) × Spec C [ I ] −→ T (cid:48) × Spec C [ I ] −→ T (cid:48) which is different from the structure of C T (cid:48) [ u ] over T (cid:48) coming from the fiber product structure. We canconsider the quotient sheaf (cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ u ]/ T (cid:48) (cid:1) hol (cid:14)(cid:0) I O hol ( C T (cid:48) \ Γ T (cid:48) )[ u ] dI (cid:1) and define a subsheaf Ω C T (cid:48) ,u of (cid:0) ι ( C T (cid:48) \ Γ T (cid:48) )[ u ] (cid:1) ∗ (cid:16)(cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ u ]/ T (cid:48) (cid:1) hol (cid:14) ( I O hol ( C T (cid:48) \ Γ T (cid:48) )[ u ] dI ) (cid:17) locally generatedby (cid:40) d ˜ z ( i ) (cid:0) ˜ z ( i ) (cid:1) m i − (cid:15) m i (cid:41) ∪ κ (cid:88) q =1 (cid:0) ι C T (cid:48) \ Γ T (cid:48) (cid:1) ∗ (cid:0) O hol C T (cid:48) \ Γ T (cid:48) (cid:1) d ¯ h q around points p ∈ (Γ ( i ) ) T (cid:48) [ u ] and locally generated by { d ˜ z } ∪ { d ¯ h j | ¯ h j ∈ I } around points p ∈ (cid:0) C T (cid:48) \ Γ T (cid:48) (cid:1) [ u ].Here ¯ h , . . . , ¯ h κ is a basis of I and z is a local holomorphic coordinate of C T (cid:48) \ Γ T (cid:48) over T (cid:48) . We denote theimage of Ω C T (cid:48) ,u ∧ Ω C T (cid:48) ,u in ( ι ( C T (cid:48) \ Γ T (cid:48) )[ u ] ) ∗ (cid:16)(cid:0) Ω C T (cid:48) \ Γ T (cid:48) )[ u ]/ T (cid:48) (cid:1) hol (cid:14)(cid:0) I O hol ( C T (cid:48) \ Γ T (cid:48) )[ u ] dI (cid:1)(cid:17) by Ω C T (cid:48) ,u .For each i = 1 , . . . , n , we consider the sheaf of differential forms Ω U i ) M (cid:48) [ u ] /M (cid:48) with respect to( U i ) M (cid:48) [ u ] (cid:44) → C × P ( M (cid:48) × Spec C [ I ]) −→ M (cid:48) × Spec C [ I ] −→ M (cid:48) , NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 73 where the last two arrows are the trivial projections. From the above projection, a ring homomorphismfrom the polynomial ring O holM (cid:48) [˜ z ( i ) ] −→ O hol ( U i ) M (cid:48) [ u ] is induced. We denote the image of a matrix A ( z ( i ) ) of polynomials in z ( i ) with coefficients in O holM (cid:48) underthis ring homomorphism by A (˜ z ( i ) ).Note that we can write u ∗ ( ν ( i ) ( T )) = ν ( i ) hor ( T ) + s (cid:88) q =1 ¯ h q ν ( i ) u,q ( T )with ν ( i ) hor ( T ) = r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) hor,l,j (˜ z ( i ) ) j T l ν ( i ) u,q ( T ) = r − (cid:88) l =0 c ( i ) u,q,l,j (˜ z ( i ) ) j T l , where c ( i ) hor,l,j and c ( i ) u,q,l,j are pullbacks of c ( i ) l,j , c ( i ) u,q,l,j ∈ O holM (cid:48) under the composition of the trivial projections( U i ) M (cid:48) [ u ] −→ M (cid:48) [ u ] −→ M (cid:48) . Definition 5.12. Under the above notation, we say that a tuple (cid:0) E u , ∇ u , {N ( i ) u } (cid:1) is a horizontal lift of (cid:0) ˜ E M (cid:48) , ˜ ∇ M (cid:48) , { ˜ N ( i ) M (cid:48) } (cid:1) with respect to u and with respect to blocks of local horizontal lifts (cid:16) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:17) if(1) E u is a rank r holomorphic vector bundle on C M (cid:48) [ u ] ,(2) ∇ u : E u −→ E u ⊗ O hol CT (cid:48) [ u ] Ω C T (cid:48) , [ u ] is a morphism of sheaves satisfying ∇ u ( f a ) = a ⊗ df + f ∇ u ( a ) for f ∈ O hol C M (cid:48) [ u ] and a ∈ E u ,(3) ∇ u is integrable in the sense that for each local expression f ...f r (cid:55)→ df ...df r + (cid:32) Ad ˜ z + κ (cid:88) l =1 B l d ¯ h l (cid:33) f ...f r of ∇ u on E u | U [ u ] ∼ = O ⊕ rU [ u ] for an open subset U [ u ] ⊂ ( C M (cid:48) \ Γ M (cid:48) )[ u ], the equality d (cid:16) A d ˜ z + κ (cid:88) l =1 B l d ¯ h l (cid:17) + (cid:16) A d ˜ z + κ (cid:88) l =1 B l d ¯ h l (cid:17) ∧ (cid:16) A d ˜ z + κ (cid:88) l =1 B l d ¯ h l (cid:17) = 0holds in Ω C T (cid:48) ,u , where { ¯ h , . . . , ¯ h κ } is a basis of I over C .(4) N ( i ) u : E u | D ( i ) M (cid:48) [ u ] −→ E u | D ( i ) M (cid:48) [ u ] is an endomorphism satisfying ϕ ( i ) µ ( N ( i ) u ) = 0,(5) the relative connection ∇ u defined by the composition ∇ u : E u ∇ u −−→ E u ⊗ Ω C T (cid:48) ,u −→ E u ⊗ Ω C M (cid:48) [ u ] /M (cid:48) [ u ] ( D M (cid:48) [ u ] ) hol satisfies ( u ∗ ν ( i ) )( N ( i ) u ) d ˜ z ( i ) (cid:0) ˜ z ( i ) (cid:1) m i − (cid:15) m i = ∇ u (cid:12)(cid:12) D ( i ) M (cid:48) [ u ] for any i ,(6) (cid:0) E u , ∇ u , {N ( i ) u } (cid:1) ⊗ O holM (cid:48) [ u ] /I O holM (cid:48) [ u ] ∼ = (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) ,(7) there is an isomorphism θ ( i ) ,u : E u (cid:12)(cid:12) ( U i ) M (cid:48) [ u ] ∼ −→ ( O hol ( U i ) M (cid:48) [ u ] ) ⊕ r which is a lift of the given isomorphism θ ( i ) | ( U i ) M (cid:48) : ˜ E | ( U i ) M (cid:48) ∼ −→ ( O hol ( U i ) M (cid:48) ) ⊕ r such that the connection matrix of ( θ ( i ) ,u ⊗ id) ◦ ∇ u ◦ ( θ ( i ) ,u ) − is given by (cid:16) A ( i ) (˜ z ( i ) , (cid:15) ) + κ (cid:88) q =1 ¯ h q r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) u,q,l,j ˜Ξ ( i ) l,j (˜ z ( i ) ) (cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i − (cid:15) m i + κ (cid:88) q =1 r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) u,q,l,j B ( i ) l,j (˜ z ( i ) ) d ¯ h q . Lemma 5.13. There exists a unique horizontal lift (cid:0) E u , ∇ u , {N ( i ) u } (cid:1) of (cid:0) ˜ E M (cid:48) , ˜ ∇ M (cid:48) , { ˜ N ( i ) M (cid:48) } (cid:1) with respect to u and with respect to blocks of local horizontal lifts (cid:16) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:17) .Proof. The proof of this lemma is the same as that of Proposition 5.11 and we omit the detail.We take the same open covering { U β } as in the proof of Proposition 5.11. We consider the connection ∇ flat P × M (cid:48) [¯ h ] ,u on ( O hol P × M (cid:48) [ u ] ) ⊕ r given by the connection matrix (cid:16) A ( i ) (˜ z ( i ) , (cid:15) ) + κ (cid:88) q =1 ¯ h q r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) u,q,l,j ˜Ξ ( i ) l,j (˜ z ( i ) ) (cid:17) d ˜ z ( i ) (˜ z ( i ) ) m i − (cid:15) m i + κ (cid:88) q =1 r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) u,q,l,j B ( i ) l,j (˜ z ( i ) ) d ¯ h q with respect to u . Let N ( i ) u be the endomorphism obtained by substituting (cid:15) m i for ( z ( i ) ) m i in ψ ( i ) (cid:16) A ( i ) (˜ z ( i ) , (cid:15) ) + κ (cid:88) q =1 ¯ h q r − (cid:88) l =0 m i − (cid:88) j =0 c ( i ) u,q,l,j ˜Ξ ( i ) l,j (˜ z ( i ) ) (cid:17) , where ψ ( i ) is given in (81). Then (cid:0) ( O hol P × M (cid:48) [ u ] ) ⊕ r , ∇ flat P × M (cid:48) [¯ h ] ,u (cid:12)(cid:12) ( U i ) M (cid:48) [ u ] , {N ( i ) u } (cid:1) becomes a local horizontallift. Patching the local horizontal lifts altogether, we obtain a unique horizontal lift in the same way asProposition 5.11. (cid:3) Proposition 5.14. The morphism T hol T ◦ / ∆ (cid:15) (cid:51) v (cid:55)→ Φ( v ) ∈ ( π ◦ ) ∗ T holM ◦ / ∆ (cid:15) defined in (88) is an O hol T ◦ -homomorphism.Proof. Take an open subset T (cid:48) ⊂ T ◦ and holomorphic vector fields v , v ∈ H (cid:0) T (cid:48) , T hol T ◦ / ∆ (cid:15) (cid:1) . Let u : T (cid:48) × Spec C [ h , h ] / ( h , h , h h ) −→ T (cid:48) be the morphism such that the restriction u | T (cid:48) × Spec C [ h i ] / ( h i ) corresponds to v i for i = 1 , 2. Apply-ing Lemma 5.13 to C [ I ] = C [ h , h ] / ( h , h h , h ), we can take a horizontal lift (cid:0) E u , ∇ u , {N ( i ) u } (cid:1) of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to u and with respect to the blocks (cid:0) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:1) of local hori-zontal lifts. We can see by construction that the restriction (cid:0) E u , ∇ u , {N ( i ) u } (cid:1)(cid:12)(cid:12) M (cid:48) × Spec C [ h i ] / ( h i ) coincideswith the horizontal lift (cid:0) E v i , ∇ v i , {N ( i ) v i } (cid:1) of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to v i . So the morphism M (cid:48) × Spec C [ h , h ] / ( h , h h , h ) −→ M α C , D (˜ ν , µ ) × B B (cid:48) determined by the flat family (cid:0) E u , ∇ u , {N ( i ) u } (cid:1) coincides with the one given by the pair (Φ( v ) , Φ( v )) ofvector fields, where ∇ u : E u −→ E u ⊗ Ω C T (cid:48) [ u ] / T (cid:48) [ u ] ( D T (cid:48) [ u ] ) hol is the relative connection induced by ∇ u .From the definition of the addition of vector fields, the restriction (Φ( v ) , Φ( v )) | M (cid:48) × Spec C [ h ,h ] / ( h − h ,h ) to the diagonal coincides with Φ( v ) + Φ( v ). On the other hand, we can see by the construction thatthe restriction (cid:0) E u , ∇ u , {N ( i ) u } (cid:1)(cid:12)(cid:12) M (cid:48) × Spec C [ h ,h ] / ( h − h ,h ) is a horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) withrespect to v + v and with respect to the blocks of local horizontal lifts (cid:0) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:1) in the sense ofProposition 5.11. So we have Φ( v + v ) = Φ( v ) + Φ( v ).Take a holomorphic function f ∈ H ( T (cid:48) , O hol T (cid:48) ) and a holomorphic vector field v ∈ H (cid:0) T (cid:48) , T hol T ◦ / ∆ (cid:15) (cid:1) . Let σ f : T (cid:48) × Spec C [ h ] / ( h ) −→ T (cid:48) × Spec C [ h ] / ( h )be the morphism corresponding to the ring homomorphism O hol T (cid:48) [ h ] / ( h ) (cid:51) a + b ¯ h (cid:55)→ a + bf ¯ h ∈ O hol T (cid:48) [ t ] / ( h )and let id × σ f : M (cid:48) × T (cid:48) T (cid:48) × Spec C [ h ] / ( h ) −→ M (cid:48) × T (cid:48) T (cid:48) × Spec C [ h ] / ( h )be its base change. If (cid:0) E v , ∇ v , {N ( i ) v } (cid:1) is a horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to v andwith respect to the blocks of local horizontal lifts (cid:0) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:1) , then we can see by the construction thatthe pull back (1 × σ f ) ∗ (cid:0) E v , ∇ v , {N ( i ) v } (cid:1) is a horizontal lift of (cid:0) ˜ E holM (cid:48) , ˜ ∇ holM (cid:48) , { ˜ N ( i ) ,holM (cid:48) } (cid:1) with respect to f v and with respect to the blocks of local horizontal lifts (cid:0) ∇ flat P × M (cid:48) [¯ h ] ,v ( i ) l,j (cid:1) . By the definition of O hol T (cid:48) -module NFOLDING OF THE UNRAMIFIED IRREGULAR SINGULAR GENERALIZED ISOMONODROMIC DEFORMATION 75 structure on the tangent bundle, we can see that the pull-back (cid:0) (id × σ f ) ∗ E v , (id × σ f ) ∗ ∇ v , { (1 × σ f ) ∗ N ( i ) v } (cid:1) of the flat family (cid:0) E v , ∇ v , {N ( i ) v } (cid:1) corresponds to f Φ( v ). So we have Φ( f v ) = f Φ( v ). Hence we have provedthat Φ is an O hol T ◦ -homomorphism. (cid:3) By the adjoint bijection(89) Hom O holM ◦ (cid:16) ( π ◦ ) ∗ T hol T ◦ / ∆ (cid:15) , T holM ◦ / ∆ (cid:15) (cid:17) ∼ = Hom O hol T ◦ (cid:16) T hol T ◦ / ∆ (cid:15) , ( π ◦ ) ∗ T holM ◦ / ∆ (cid:15) (cid:17) , the O hol T ◦ -homomorphism Φ : T hol T ◦ / ∆ (cid:15) −→ ( π ◦ ) ∗ T holM ◦ / ∆ (cid:15) given in (88) corresponds to an O holM ◦ -homomorphismΨ : ( π ◦ ) ∗ T hol T ◦ / ∆ (cid:15) −→ T holM ◦ / ∆ (cid:15) . Since Φ satisfies dπ ◦ ◦ Φ( v ) = v for vector fields v ∈ T hol T ◦ / ∆ (cid:15) , the homo-morphism Ψ is a splitting of the surjection T holM ◦ / ∆ (cid:15) dπ ◦ −−→ ( π ◦ ) ∗ T hol T ◦ / ∆ (cid:15) canonically induced by the smoothmorphism π ◦ : M ◦ −→ T ◦ . Furthermore we can see Ψ (cid:12)(cid:12) M α C , D (˜ ν , λ ) (cid:15) =0 ∩ M ◦ = Ψ (cid:12)(cid:12) M α C , D (˜ ν , λ ) (cid:15) =0 ∩ M ◦ from itsconstruction. Thus we have proved Theorem 0.1. Example 5.15. Let us consider the case of g = 0, r = 2, n = 2, m = 2, m = 1 and a = deg E = 0. So C = P , D (1) = { z − (cid:15) = 0 } and we may assume D (2) = {∞} . We choose z (1) = z and z (2) = w = 1 /z .We take the exponent ν so generic that res z = ∞ (cid:18) ν (1) ( µ k ) dzz − (cid:15) (cid:19) + res w = ∞ (cid:18) ν (2) ( µ k ) dww (cid:19) / ∈ Z forany choice of k , k ∈ { , } . Then the ( ν , µ ) connections are irreducible and correspond to the classicalhypergeometric equations. The moduli space M α P ,D (˜ ν , µ ) consists of a single point because of the rigidityof the hypergeometric equations. For a ( ν , µ )-connection ( E, ∇ , { N ( i ) } ) ∈ M α P ,D (˜ ν , µ ), we have E ∼ = O ⊕ P and ∇| U is given by a connection matrix(90) A ( (cid:15) ) + A ( (cid:15) ) zz − (cid:15) dz. The above connection matrix is uniquely determined by ( E, ∇ ) up to a constant conjugate and the matricesΞ (1) l,j ( z ) ( l = 0 , j = 0 , 1) given in (80) are systematically determined. We writeΞ (1) l,j ( z ) = C (1) l,j, ( (cid:15) ) + C (1) l,j, ( (cid:15) ) z. If we take an adjusting data (cid:0) R (1) l,j, , R (1) l,j, (cid:1) , we have C (1) l,j, ( (cid:15) ) = (cid:2) A , R (1) l,j, (cid:3) + (cid:2) A , R (1) l,j, (cid:3) and we define˜Ξ (1) l,j ( z ) = C (1) l,j, − (cid:2) A , R (1) l,j, (cid:3) − (cid:15) (cid:2) A , R (1) l,j, (cid:3) . There is an ambiguity in the choice of adjusting data (cid:0) R (1) l,j, , R (1) l,j, (cid:1) . If (cid:0) R (cid:48) (1) l,j, , R (cid:48) (1) l,j, (cid:1) is another one,then C (1) l,j, = (cid:2) A , R (1) l,j (cid:3) + (cid:2) A , R (1) l,j, (cid:3) = (cid:2) A , R (cid:48) (1) l,j, (cid:3) + (cid:2) A , R (cid:48) (1) l,j, (cid:3) . Since we are choosing A , A generic,the full matrix ring is generated by A , A , [ A , A ] , I . Furthermore, im ad( A ) ∩ im ad( A ) is generatedby [ A , A ]. Since (cid:2) A , R (1) l,j, − R (cid:48) (1) l,j, (cid:3) = − (cid:2) A , R (1) l,j, − R (cid:48) (1) l,j, (cid:3) ∈ im ad( A ) ∩ im ad( A ), we can write R (1) l,j, − R (cid:48) (1) l,j, = aA + bA and R (1) l,j, − R (cid:48) (1) l,j, = cA + aA for some functions a, b, c defined on an opensubset of the moduli space M α P ,D (˜ ν , µ ). If we put ˜Ξ (cid:48) (1) l,j ( z ) := C (1) l,j, − (cid:2) A , R (cid:48) (1) l,j, (cid:3) − (cid:15) (cid:2) A , R (cid:48) (1) l,j, (cid:3) , then˜Ξ (1) l,j ( z ) − ˜Ξ (cid:48) (1) l,j ( z ) = (cid:2) A , R (1) l,j, − R (cid:48) (1) l,j, (cid:3) − (cid:15) (cid:2) A , R (1) l,j, − R (cid:48) (1) l,j, (cid:3) = ( b − (cid:15) c ) (cid:2) A , A (cid:3) . So we have (cid:0) I − ¯ h ( b − (cid:15) c ) A (cid:1) − A + A z + ¯ h ˜Ξ (1) l,j ( z ) z − (cid:15) dz (cid:0) I − ¯ h ( b − (cid:15) c ) A (cid:1) = A + A z + ¯ h ˜Ξ (cid:48) (1) l,j ( z ) z − (cid:15) dz which means that there is no essential ambiguity in the relative connection given by the connection matrix A + A z + ¯ h ˜Ξ (1) l,j ( z ) z − (cid:15) dz. up to a global automorphism. However, there is an ambiguity in the choice of B (1) l,j such that the connectionmatrix A ( (cid:15) ) + A ( (cid:15) ) z + ¯ h ˜Ξ (1) l,j ( z ) z − (cid:15) dz + B (1) l,j ( z ) d ¯ h gives a horizontal lift. Indeed, for a fundamental solution Y ∞ ( z, (cid:15) ) of ∇ near ∞ , there is an ambiguity in Y ∞ ( z, (cid:15) ) + ¯ hB (1) l,j ( z ) Y ∞ ( z, (cid:15) ) by an action of ( I + ¯ h ( c I + c Mon ∞ )) from the right with c ≡ , c ≡ (cid:15) ), where Mon ∞ is the monodromy matrix of Y ∞ ( z, (cid:15) ) along a loop around ∞ . If we write Y ∞ ( z, (cid:15) ) + ¯ hB (1) l,j ( z ) Y ∞ ( z, (cid:15) ) = (˜ y , ˜ y ) with ˜ y , ˜ y two independent hypergeometric solutions, then the ambiguity isessentially given by a replacement of (˜ y , ˜ y ) with ((1 + ¯ hb )˜ y , (1 + ¯ hb )˜ y ), where b ≡ , b ≡ (cid:15) ).Notice that we can in fact assume c = 0 after a normalization via applying a global automorphism, butthere is still an ambiguity arising from ¯ hc . Acknowledgment. The author would like to thank Professor Takeshi Abe for giving him opportunitiesto have discussions on the subject in this paper. He also would like to thank Professor Hironobu Kimura andProfessor Yoshishige Haraoka for giving him valuable comments when the author tried to start the studyof the subject in this paper. The author would like to thank Professor Indranil Biswas and Professor Bala-subramanian Aswin for the hospitality at the conference “Quantum Fields, Geometry and RepresentationTheory” held at Bangalore in 2018. He also thanks Professor Takuro Mochizuki and Professor Masa-HikoSaito who had discussion with him related to this paper. The author would like to thank Professors Hi-royuki Inou and Masayuki Asaoka for telling him elementary textbooks for an introduction to dynamicalsystems, though the author have not followed most of them yet. The author would like to thank ProfessorIndranil Biswas for the hospitality at the conference “Bundles -2019” at Mumbai in 2019. 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